The 10th Pacific RIM Geometry Conference 2011 Osaka-Fukuoka: Part I
| Dec. 1 (Thu) | Morning Session | |
| 9:40-9:45 | Opening Remarks | |
| 9:50-10:50 | Shu-Cheng Chang (National Taiwan University, Taiwan, ROC) | |
| “Li-Yau gradient estimate and entropy formulae for the CR heat equation in a closed pseudohermitian 3-manifold” | ||
| Abstract: In this paper, we derive two sub-gradient estimates of the CR heat equation in a closed pseudohermitian 3-manifold which are served as the CR version of Li-Yau gradient estimate. With its applications, we first get a subgradient estimate of logarithm of the positive solution of CR heat equation. Secondly, we have the Harnack inequality and upper bound estimate for the heat kernel. Finally, we obtain Perelman-type entropy formulae for the CR heat equation. | ||
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| 11:00-12:00 | Carlos Olmos (National University of Cordoba, Argentina) | |
| “Killing fields, holonomy and the index of symmetry” | ||
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Abstract:
This talk is mainly based on a work, still in preparation, with Silvio Reggiani.
We would like to draw the attention to some concept that we call the
index of symmetry 0 ≤ is(M) ≤ n of a Riemannian manifold Mn.
The index of symmetry can be defined as the dimension of the tangent subspace where any natural
Riemannian tensor is parallel (or, equivalently, the dimension of the space of Killing
fields that are parallel at a given point). One has that M is symmetric if and only
if is(M) = n We are, of course, interested on non-symmetric spaces with positive
index of symmetry. In this case one can prove that is(M) ≤ n − 2 (in other words,
the co-index of symmetry is at least 2, for a non-symmetric space). We have some
general results and many questions.
Many examples of spaces with non-trivial index of symmetry arise from naturally
reductive spaces (we will also refer to a previous joint work with Reggiani related
to naturally reductive spaces and holonomy, Crelle’s 2011).
Also the unit tangent bundle over the sphere Sn of curvature 2 has is(Sn) = n − 1.
We prove the following result Theorem Let Mn be a compact locally irreducible homogeneous Riemannian manifold which is not locally symmetric. Let k := n−is(M) be its co-index of symmetry. Then there is a subgroup of isometries G ⊂ I(M), which acts transitively on M and such that dim(G) ≤ (1/2)k(k + 1). Moreover, if the equality holds, then, up to a cover, G = Spin(k + 1) and G has non trivial isotropy, if k ≥ 3. This allows us to classify the homogeneous spaces with low co-index of symmetry. For instance the spaces with co-index of symmetry 2 correspond to two distinguished families of one-parameter left invariant metrics on Spin(3). It is an interesting fact that there is a nice equivariant “Gauss map” from a homogeneous space M with non-trivial index of symmetry, into an appropriate Grassmannian. The subjects of this talk may be regarded as an effort to explore Riemannian manifolds that are symmetric up to some defect (in the hope of finding distinguished non-symmetric homogeneous manifolds). In some sense, our philosophy is in the direction of the concept of co-polarity by Claudio Gorodski, that measures how a representation, orbit like, differ from a symmetric (isotropy) representation. |
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| Dec. 1 (Thu) | Parallel Session (A) | |
| 13:30-14:20 | Yaroslav V. Bazaikin (Sobolev Institute of Mathematics, Novosibirsk, Russia) | |
| “On G2-holonomy metrics based on S3 × S3” | ||
| Abstract: We discuss one-parameter family of complete G2-holonomy Riemannian metrics obtained by deformation of standard cone metric over S3 × S3. | ||
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| 14:30-15:20 | Nickolai Erokhovets (Moscow State University, Russia) | |
| “Towards the Theory of Buchstaber Invariant” | ||
| Abstract: Toric topology associates to each simple n-polytope P with m faces the smooth moment-angle manifold ZP with the canonical Tm action such that ZP /Tm = P. This gives a way to study the combinatorics of P in terms of the topology of ZP and vice versa. This idea is realized by the Buchstaber invariant s(P) – the combinatorial invariant of simple polytope P equal to the maximal dimension of torus subgroups H ≃ Tk ⊆ Tm acting freely on ZP – which is in some sense a measure of symmetry of ZP. It can be shown that 1 ≤ s(P) ≤ m − n. In 2002 Victor M. Buchstaber stated a problem to find an effective method to calculate s(P) in terms of the combinatorics of P. The Buchstaber invariant has been studied since 2001. Nowadays there are quite enough general results about s(P) to look towards the theory of Buchstaber invariant. For example, we will show that s(P) = 1 iff P = Δn; for any k ≥ 2 there exists a polytope P with m−n = k and s(P) = 2; s(P) can not be calculated if only the f-vector and the chromatic number γ(P) are known. We will also show the behavior of the Buchstaber invariant under constructions of the polytope theory and it’s connection with classical and modern combinatorial invariants of polytopes. | ||
| 16:00-16:30 | Imsoon Jeong*, Seonhui Kim and Young Jin Suh (Kyungpook National University, Korea) (*: speaker) | |
| “Real hypersurfaces in complex two-plane Grassmannians with ξ-parallel structure Jacobi operator” | ||
| Abstract: In this talk we give a characterization of Hopf hypersurfaces of Type (A) in complex two-plane Grassmannians G2(Cm+2), that is, a tube over a totally geodesic G2(Cm+1) in G2(Cm+2) with ξ-parallel structure Jacobi operator. | ||
| 16:30-17:00 | Imsoon Jeong, Hyunjin Lee* and Young Jin Suh (Kyungpook National University, Korea) (*: speaker) | |
| “Characterizations of Hopf hypersurfaces in complex two-plane Grassmannians related to generalized Tanaka-Webster connection” | ||
| Abstract: In this talk, we introduce the notion of generalized Tanaka-Webster connection (in short, g-Tanaka-Webster connection) for hypersurfaces in complex two-plane Grassmannians G2(Cm+2). Moreover, we consider various parallelisms of shape operator with respect to g-Tanaka-Webster connection, namely, g-Tanaka-Webster parallel, g-Tanaka-Webster ξ-parallel, g-Tanaka-Webster ξ-parallel, and g-Tanaka-Webster D⊥-parallel. By using there concepts, we give some characterizations of Hopf hypersurfaces in G2(Cm+2). | ||
| Dec. 1 (Thu) | Parallel Session (B) | |
| 13:30-14:20 | Makiko Sumi Tanaka (Tokyo University of Science, Japan) | |
| “Antipodal sets of compact Riemannian symmetric spaces and their applications” | ||
| Abstract: This talk is based on a joint work with Hiroyuki Tasaki. We investigate fundamental properties of antipodal sets of symmetric R-spaces. We also investigate the intersection of two real forms in a Hermitian symmetric space of compact type and obtained that the intersection is an antipodal set, and moreover, we obtained that its cardinality is equal to the 2-number of the real form if two real forms are congruent. As a consequence we obtained that every real form of a Hermitian symmetric space of compact type is a globally tight Lagrangian submanifold. | ||
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| 14:30-15:20 | Adela Mihai (University of Bucharest, Romania) | |
| “Normal Complex Contact Metric Manifolds” | ||
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Abstract:
In this lecture the complex contact manifolds from a Riemannian geometric
point of view, comparing the ideas with those of real contact metric geometry,
are discussed. One important notion is that of a normal complex contact metric
structure. In the first part, I will present the recent work (D. E. Blair, A. Mihai) on locally symmetric normal complex contact metric manifolds along with the role played by reflections in the integral submanifolds of the vertical subbundle. Also, the properties of homogeneity and local symmetry of complex (k, μ)-spaces are shown. The second part consists in recent definitions and studies of submanifolds of complex contact metric manifolds. References: D. E. Blair, A. Mihai, Symmetry in complex contact geometry, Rocky Mount. J. Math., to appear. D. E. Blair, A. Mihai, Homogeneity and local symmetry of complex (k, μ)-spaces, Israel J. Math., DOI: 10.1007/s11856-011-0089-2. |
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| 16:00-16:30 | Xianfeng Wang (Nankai University, P. R. China) | |
| “Lagrangian submanifolds in complex projective space with parallel second fundamental form” | ||
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Abstract:
From the Riemannian geometric point of view, one of the most fundamental
problems in the study of Lagrangian submanifolds is the classification
of Lagrangian submanifolds with parallel second fundamental form. In 1980’s, H.
Naitoh classified the Lagrangian submanifolds with parallel second fundamental form
and without Euclidean factor in complex projective space, by using the theory of
Lie groups and symmetric spaces. He showed that such a submanifold is always
locally symmetric and is one of the symmetric spaces: SO(k + 1)/SO(k) (k ≥ 2),
SU(k)/SO(k) (k ≥ 3), SU(k) (k ≥ 3), SU(2k)/Sp(k) (k ≥ 3), E6/F4. In this paper, we completely classify the Lagrangian submanifolds in complex projective space with parallel second fundamental form by an elementary geometrical method. We prove that such a Lagrangian submanifold is either totally geodesic or the Calabi product of a point with a lower dimensional Lagrangian submanifold with parallel second fundamental form, or the Calabi product of two lower dimensional Lagrangian submanifolds with parallel second fundamental form, or one of the standard symmetric spaces: SO(k + 1)/SO(k) (k ≥ 2), SU(k)/SO(k) (k ≥ 3), SU(k) (k ≥ 3), SU(2k)/Sp(k) (k ≥ 3), E6/F4. This is joint work with Professor Franki Dillen, Professor Haizhong Li and Professor Luc Vrancken. |
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| 16:30-17:00 | Seonhui Kim*, Hyunjin Lee and Young Jin Suh (Kyungpook National University, Korea) (*: speaker) | |
| “A new condition of real hypersurfaces in complex two-plane Grassmannians” | ||
| Abstract: We give a characterization of Hopf hypersurfaces of Type (A), that is, a tube over a totally geodesic G2(Cm+1) in complex two plane Grassmannians G2(Cm+2) in terms of commuting condition φφ1A = Aφ1φ between the shape operator A and the structure tensors φ, φ1 for real hypersurfaces in G2(Cm+2). | ||
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| Dec. 2 (Fri) | Morning Session | |
| 9:50-10:50 | Victor M. Buchstaber (Steklov Mathematical Institute & Moscow State University, Russia) | |
| “Symplectic nilmanifolds and applications” | ||
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Abstract:
The talk will be devoted to the remarkable sequence of bundles Mn →
Mn−1, n ≥ 1, with fiber the circle. Each Mn is a smooth nilmanifold with a 2-form
for n ≥ 2, which gives a symplectic structure on M2k, and a contact structure on M2k+1. This sequence plays an important role in diffenent areas of mathematics. We will discuss the differential-geometric and algebro-topologic results and unsolved questions, concerning this sequence. |
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| 11:00-12:00 | Yael Karshon (Toronto University, Canada) | |
| “Counting toric actions” | ||
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Abstract:
In how many different ways can a two-torus act on a given simply
connected symplectic four-manifold? If the second Betti number is one or two, the
answer has been known for a while. For a higher Betti number, our (“soft”) proof
that there are only finitely many inequivalent torus actions did not enable us to
count these actions. I will report on recent work, in which we reduce this counting question to combinatorics by expressing the manifold as a symplectic blowup in a way that is compatible with all the torus actions simultaneously. For this we use the theory of pseudoholomorphic curves. This work is joint with Liat Kessler and Martin Pinsonnault. |
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| Dec. 2 (Fri) | Afternoon Session | |
| 13:30-14:20 | Dmitry V. Gugnin (Moscow State University, Russia) | |
| “Smith-Dold Branched Coverings and Cup-Length” | ||
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Abstract:
Smith-Dold branched coverings are finite-fold branched coverings of
Hausdorff spaces of a special type. They were defined by L.Smith in 1983 as
a generalization of unbranched finite-fold coverings on which can be extended a
(co)homology transfer. A.Dold gave a characterization of such a coverings in terms
of actions of finite groups on topological spaces. Subsequently branched coverings
of such a type were called Smith-Dold branched coverings. There are at least 3 important for topology classes of maps which are n-fold Smith-Dold branched coverings: (1) unbranched n-fold coverings of Hausdorff spaces. (2) the projection map f : X → X/G on the quotient space of X by an action of a group G of order n. (3) usual n-fold branched coverings of PL (smooth) manifolds. In item (3) (the PL case) branched coverings of manifolds mean open-closed PL finite-fold maps of connected PL manifolds. Using the cohomology transfer due to L. Smith it is easy to show that for any n-fold Smith-Dold branched covering f : X → Y the induced homomorphism of rational and Zp, p > n, cohomology rings is a monomorphism. Therefore if a space X has at least one Betti number (rational or Zp, p > n) bq(X) which is less than the corresponding Betti number bq(Y) of a space Y, then there is no n-fold branched covering f : X → Y. So there is the following easy principle: (Easy P.) A simple space cannot cover (with an arbitrarily large n) a complicated space. One of the most common numerical invariants of a cohomology ring of a space is a cup-length. Cup-length lR(X) of a space X is the maximal number of homogeneous elements of positive degrees of a cohomology ring H∗(X;R) which when multiplied give a nonzero product (R is an arbitrary commutative coefficient ring). Cup-length is the lower bound for Lusternik-Schnirelmann category of a space. Denote by l(X) the rational cup-length of a space X, and by lp(X) − − − Zp cup-length of X. I. Berstein and A. L. Edmonds in 1978 proved the following Theorem. For any n-fold branched covering f : Xm → Ym of connected PL (Top) orientable closed manifolds the following inequality holds: l(Ym) ≥ l(Xm)/n. This result of I. Berstein and A. L. Edmonds made a good start for the following “hard”principle. (“Hard”P.) A too complicated space cannot cover a too simple space with a relatively small number of sheets n. The main result to be presented on the talk is the following general inequality slightly more weaker than Berstein-Edmonds inequality. Theorem 1. For any n-fold Smith-Dold branched covering f : X → Y of locally contractible paracompact spaces the following inequality holds: l(Y) + 1 ≥ (l(X) + 1)/n; lp(Y) + 1 ≥ (lp(X) + 1)/n, ∀p > n.
The second result extends the original Berstein-Edmonds inequality:Theorem 2. For any n-fold Smith-Dold branched covering f : Xm → Ym, with the base — connected top. orientable closed manifold, and the ENR (euclidean neighborhood retract) total space, the following inequality holds: l(Ym) ≥ l(Xm)/n; lp(Ym) ≥ lp(Xm)/n, ∀p > n.
Our techniques is quite different from the one of I. Berstein and A. L. Edmonds, and
is based on our extension of V. M. Buchstaber and E. G. Rees theory of Frobenius n-homomorphisms
to graded algebras. Moreover, by the Berstein-Edmonds technics
it cannot be proven even Theorem 2, because their approach needs the Poincare
Duality of the total space, which does not hold for arbitrary ENR’s (arbitrary pseudomanifolds).
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| 14:30-15:20 | Megumi Harada (McMaster University, Canada) | |
| “An invitation to (Newton-)Okounkov bodies” | ||
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Abstract:
This is not a standard research talk. It is not even a standard expository
talk. Instead, it is an invitation to, and an advertisement of, a new and rapidly
developing research area at the intersection of algebraic geometry, symplectic geometry,
representation theory, and combinatorics. My main goal is to set the stage,
to illustrate (some of) the connections between Okounkov bodies and the above-mentioned
research areas, and (time permitting) to outline a small sample of the
many open questions in the field. The celebrated Bernstein-Kushnirenko theorem from Newton polyhedra theory relates the number of solutions of a system of polynomial equations with the volumes of their corresponding Newton polytopes. This motivated developments in the theory of toric varieties, which connects the combinatorics of a convex integral polytope Δ with the (equivariant) geometry of the associated toric variety X(Δ). In the more general setting of symplectic manifolds and Hamiltonian actions, the Atiyah/Guillemin-Sternberg and Kirwan convexity theorems link equivariant symplectic and algebraic geometry to the combinatorics of moment map polytopes. In the case of a toric variety X(Δ), the moment map polytope Δ fully encodes the geometry of X(Δ), but this fails in general. In ground-breaking work, Okounkov constructs, for an (irreducible) projective variety X ⊆ P(V ) equipped with an action of a reductive algebraic group G, a convex body Δ˜ and a natural projection from Δ˜ to the moment map polytope Δ of X. The volumes of the fibers of this projection encode the so-called Duistermaat-Heckman measure, and in particular, one recovers the degree of X (i.e. the symplectic volume) from Δ˜. Recently, Kaveh-Khovanskii and Lazarsfeld-Mustata have vastly generalized Okounkov’s ideas; specifically, given the data of a variety X and a (big) divisor D on X, they construct a convex body Δ˜(X,D) with dimR(Δ˜(X,D)) = dimC(X) (called a Newton-Okounkov body or Okounkov body) even without presence of any group action. Thus the constructions of Kaveh-Khovanskii and Lazarsfeld-Mustata show that there are combinatorial objects of ‘maximal’dimension associated to X in great generality. As a first application, Kaveh-Khovanskii use this to prove a far-reaching generalization of the Bernstein-Kushnirenko theorem to arbitrary varieties which relates the self-intersection number of a divisor with the volume of the corresponding Newton-Okounkov body. This theory is still in its infancy and the subject is wide open. The fundamental question is: What (asymptotic) geometric data of (X,D) do the combinatorics of Okounkov bodies encode, and how? |
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| 15:40-16:30 | Qun Chen (Wuhan University, P. R. China) | |
| “The maximum principle and the Dirichlet problem for Dirac-harmonic maps” | ||
| Abstract: In this talk, we will introduce a recent work joint with J. Jost and G. F. Wang on Dirac-harmonic maps, which satisfy a system of equations consisting of a second order elliptic system and a Dirac equation on Riemannian spin manifolds. We first give a maximum principle for Dirac-harmonic maps from a Riemannian spin manifold with boundary into a regular ball in any Riemannian manifold. Then we establish a general existence theorem for boundary value problems of Dirac-harmonic maps. | ||
| 16:40-17:30 | Yuxin Dong (Fudan University, P. R. China) | |
| “Monotonicity Formulae and Holomorphicity of Harmonic Maps between Kähler manifolds” | ||
| Abstract: In this work, we introduce the stress-energy tensors of the partial energies E′(f) and E′′(f) of maps between Kaehler manifolds. Assuming the domain manifolds poss some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps into any Kaehler manifolds and harmonic maps into Kaehler manifolds with strongly semi-negative curvature respectively. These monotonicity inequalities enable us to derive some holomorphicity and Liouville type results for these pluriharmonic maps and harmonic maps. We also use the stress-energy tensors to investigate the holomorphic extension problem of CR maps. | ||
| Dec. 3 (Sat) | Morning Session | |
| 9:50-10:50 | Kaoru Ono (Hokkaido University, Japan) | |
| “Lagrangian Floer theory on compact toric manifolds” | ||
| Abstract: I will present a review on Lagrangian Floer theory on compact toric manifolds based on my joint works with K. Fukaya, Y.-G. Oh, H. Ohta. If time allows, I will also discuss applications to Calabi quasi-morphisms on the universal covering group of Hamiltonian diffeomorphisms, symplectic partial quasi-states, which are discovered by Entov and Polterovich, etc. | ||
| 11:00-12:00 | Yong-Geun Oh (University of Wisconsin-Madison, USA) | |
| “Localization of Floer homology in C0 hamiltonian topology” | ||
| Abstract: Localization of Floer homology was first introduced by Floer for a C2-small Hamiltonian H, which was the employed by the present speaker in the study of topology of Lagrangian submanifolds. In this talk, we will explain how this localization process can be carried out for a Hamiltonian that is small both in C0 topology of Hamiltonian flows and in Hofer topology of Hamiltonians, i.e., small in the sense of hamiltonian topology that is introduced by Mueller and the speaker. We will also explain how this localization gives rise to a comparison result between a ‘global’spectral invariant and a ‘local’spectral invariant. This comparison result plays a crucial role in the author’s recent study of nonsimpleness question of area-preserving homeomorphism group of the two disc. | ||
| Dec. 3 (Sat) | Afternoon Session | |
| 13:30-14:20 | Hông Vân Lê (Institute of Mathematics of ASCR, Czech Republic) | |
| “Twisted symplectic manifolds and the associated cohomologies” | ||
| Abstract: This is my joint work with Jiri Vanzura and Alexandre Vinogradov. We introduce the notion of a twisted symplectic manifold (M2n,L,∇,ω). Locally conformally symplectic manifolds are particular cases of twisted symplectic manifolds when L is the trivial line bundle. We associate with a twisted symplectic manifold (M2m,L,∇,ω) cohomology groups using the Lepage-Lefschetz decomposition. We study the relation between these new invariants, using and extending the technique of spectral sequences developed by Di Pietro and Vinogradov for symplectic manifolds. We discuss related results by many peoples, e.g. Bouche, Lychagin, Rumin, Tseng-Yau, in light of our spectral sequences. We calculate the associated cohomologies of a (2n+2)-dimensional locally conformally symplectic nilmanifold as well as those of a solvable 4-manifold. Using our theory we show an explicit example of a coorientation preserving contactomorphism of a connected contact 3-manifold, which is not isotopic to the identity through contactomorphisms. | ||
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| 14:30-15:20 | Hiroshi Iriyeh (Tokyo Denki University, Japan) | |
| “Floer homology and Hamiltonian volume minimizing properties of real forms of complex hyperquadric” | ||
| Abstract: In this talk we first calculate the Lagrangian Floer homology HF(L0,L1:Z2) of a pair of real forms (L0,L1) in the complex hyperquadric Qn(C) in the case where L0 is not necessarily congruent to L1. This yields a generalization of the Arnold-Givental inequality. Then we obtain a volume estimate for all real forms of Qn(C) under Hamiltonian isotopies, combining the inequality and a Crofton type formula obtained by Lê Hông Vân. In particular, we prove that the totally geodesic Lagrangian sphere in the complex hyperquadric is globally volume minimizing under Hamiltonian deformations. This talk is based on a joint work with Takashi Sakai and Hiroyuki Tasaki. | ||
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| Dec. 3 (Sat) | Parallel Session (A) | |
| 15:50-16:20 | Toru Kajigaya (Tohoku University, Japan) | |
| “Legendrian minimal submanifolds in Sasakian manifolds and its stability” | ||
| Abstract: Y. G. Oh introduced the notion of Hamiltonian-minimal Lagrangian submanifolds in Kähler manifolds, and his main concern is stability. An odd-dimensional version of Kähler manifolds would be Sasakian manifolds in the contact geometry. Corresponding to Hamiltonian minimal Lagrangian submanifolds, we introduce the notion of Legendrian-minimal Legendrian submanifolds in Sasakian manifolds, and investigate the stability. | ||
| 16:30-17:20 | Hui Ma (Tsinghua University, P. R. China) | |
| “On Lagrangian submanifolds in complex hyperquadrics and Hamiltonian volume variational problem” | ||
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Abstract:
We will give a short survey on Hamiltonian volume variational problem
related to Lagrangian submanifolds in Kähler manifolds. Then we will mainly
discuss the properties of compact minimal Lagrangian submanifolds embedded in a
complex hyperquadric obtained as the Gauss images of isoparametric hypersurfaces
in a sphere. Our main results are as follows: (a) The Gauss image is a monotone and cyclic Lagrangian submanifold in a complex hyperquadric with minimal Maslov number 2n/g, where g denotes the number of distinct principal curvatures of the isoparametric hypersurface. (b) The classification of homogeneous Lagrangian submanfolds. (c) The determination of the (strictly) Hamiltonian stability of the Gauss images of all compact homogeneous isoparametric hypersurfaces in spheres, by harmonic analysis on homogeneous spaces and fibrations on homogeneous isoparametric hypersurfaces. This talk is mainly based on the joint work with Professor Yoshihiro Ohnita. |
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| 17:30-18:20 | Taras Panov (Moscow State University, Russia) | |
| “Intersections of quadrics and H-minimal Lagrangian submanifolds” | ||
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Abstract:
We study the topology of Hamiltonian-minimal Lagrangian submanifolds
N in Cm constructed from intersections of real quadrics in the work of the first author.
This construction is linked via an embedding criterion to the well-known Delzant construction of Hamiltonian toric manifolds. We establish the following topological properties of N: every N embeds as a submanifold in the corresponding moment-angle manifold Z, and every N is the total space of two different fibrations, one over a torus with fibre a real moment-angle manifold R, and another over a quotient of R by a finite group (known as a small cover) with fibre a torus. These properties are used to produce new examples of H-minimal Lagrangian submanifolds with quite complicated topology. The interpretation of our construction in terms of symplectic reduction leads to its generalisation providing new examples of H-minimal submanifolds in toric varieties. The talk is based on a joint work with Andrey Mironov. Reference: Andrey Mironov and Taras Panov. Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings. Preprint(2011); arXiv:1103.4970. |
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| Dec. 3 (Sat) | Parallel Session (B) | |
| 15:50-16:20 | Peng Wang (Tongji University, P. R. China) | |
| “Willmore two-spheres in Sn+2 via Loop Group Theory” | ||
| Abstract: We consider the harmonic conformal Gauss maps of Willmore surfaces by use of loop group methods. First, we derive a generic description of the normalized potential of a Willmore harmonic map into SO(1,n+3)/SO(1,3)×SO(n). Then we consider such harmonic maps of finite uniton, via the DPW version of the theory of Burstall-Guest on harmonic maps of finite uniton. Then we give a classification of the normalized potential of Willmore harmonic maps of finite uniton. As an application, we derive a totally isotropic Willmore sphere in S6, which is not S-Willmore. | ||
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| 16:30-17:20 | Xiang Ma (Peking University, P. R. China) | |
| “The global geometry of stationary surfaces in 4-dimensional Lorentz space” | ||
| Abstract: We study the global geometry of complete stationary surfaces (i.e. zero mean curvature and space-like surfaces) in 4-dimensional Lorentz space based on a Weierstrass type representation of them. We find a series of examples with finite total curvature whose Gauss maps could not be extended to one end. We also generalize the construction of catenoid, k-noids and Enneper surface, all of them being embedded. These phenomena differ greatly with the classical minimal surfaces in 3-dimensional Euclidean space. We will also report our work on Gauss-Bonnet type theorems and the exceptional value problem about the Gauss maps. (This is a joint work with Zhiyu Liu, Changping Wang and Peng Wang.) | ||
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| 17:30-18:20 | Emma Carberry (University of Sydney, Australia) | |
| “Harmonic maps, Toda frames and extended Dynkin diagrams” | ||
| Abstract: I shall discuss harmonic maps from surfaces into homogeneous spaces G/T where G is any simple real Lie group (not necessarily compact) and T is a Cartan subgroup. All immersions of a genus one surface into G/T possessing a Toda frame can be constructed by integrating a pair of commuting vector fields on a finite dimensional Lie algebra. I will provide necessary and sufficient conditions for the existence of a Toda frame and describe those G/T to which the theory applies in terms of involutions of extended Dynkin diagrams. Applications will be given to harmonic maps into de Sitter spaces and to Willmore tori in S3. This is joint work with Katharine Turner (University of Chicago). | ||
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| Dec. 4 (Sun) | Morning Session | |
| 9:50-10:50 | Xiaobo Liu (University of Notre Dame, USA and Peking University, P. R. China) | |
| “Universal Equations for Gromov-Witten Invariants” | ||
| Abstract: There is a class of differential equations which holds for generating functions of Gromov-Witten invariants of all compact symplectic manifolds. Such equations are called universal equations. Universal equations can be used to compute Gromov-Witten invariants. They also play important roles in the study of the Virasoro conjecture. It is well known that relations in tautological ring of moduli spaces of stable curves can produce universal equations, not only for Gromov-Witten invariants, but also for any cohomological field theory which satisfies the splitting principle. A typical example of such an equation is the WDVV equation, which is a genus-0 equation and gives the associativity of the quantum cohomology. Finding such relations in higher genera is a very difficult problem. Mumford, Getzler, Belorousski-Pandharipande have found some universal equations of genus-1 and genus-2. Together with Takashi Kimura, we obtained two genus-3 universal equations. I will also talk about some topological recursion relations for all genera which was proved in a joint paper with R. Pandharipande. Some of these relations can be used to prove a conjecture of Kefeng Liu and Hao Xu. | ||
| 11:00-12:00 | Hiroshi Iritani (Kyoto University, Japan) | |
| “Quantum cohomology and periods” | ||
| Abstract: Mirror symmetry predicts that quantum cohomology of a given manifold can be calculated by periods of the mirror manifold. In this talk, I will explain that a vector bundle on the original manifold should correspond to an integration cycle of mirror periods. In this correspondence, we need a transcendental characteristic class, called the Gamma class. Conjecturally, vector bundles and Gamma class should define a pure and polarized Hodge structure on the quantum cohomology globally. I will also explain its application to the functoriality of quantum cohomology. | ||
| Dec. 4 (Sun) | Afternoon Session | |
| 13:30-14:20 | Siu-Cheong Lau (IPMU, Japan) | |
| “SYZ and mirror maps for semi-Fano toric manifolds” | ||
| Abstract: In this talk I will explain my recent joint work with K. W. Chan, N. C. Leung and H. H. Tseng on mirror maps via the SYZ approach. We derive an open analog of closed-string mirror symmetry for a class of toric manifolds, which leads to a computational method of open Gromov-Witten invariants in the toric cases via mirror symmetry. | ||
| 14:30-15:20 | Jianxun Hu (Sun Yat-sen University, P. R. China) | |
| “Degeneration formulae and its applications to local GW and DT invariants” | ||
| Abstract: Degeneration formula is one of the most important techniques in the Gromov-Witten and Donaldson-Thomas theory. In this talk, I will first introduce the degeneration formulae and then talk about how to use the degeneration technique to study the change of Gromov-Witten and Donaldson-Thomas invariants of local surfaces under blowing up along points. | ||
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| 15:40-16:30 | Chin-Lung Wang (National Taiwan University, Taiwan, ROC) | |
| “Quantum Leray-Hirsch” | ||
| Abstract: Let X be a split toric bundle over a smooth base S. I will explain how to construct the Dubrovin connection on X in terms of the Dubrovin connection on S and the Picard-Fuchs system associated to the toric fiber. The construction is natural in the sense that under an ordinary flop over S we get analytic continuations of quantum cohomology. This is a joint work with Y. P. Lee and H. W. Lin. | ||
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| 16:40-17:30 | Mohammad Ghomi (Georgia Institute of Technology, USA) | |
| “Tangent lines, inflection points, and vertices of closed curves” | ||
| Abstract: We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P + I) + V > 5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow, is based on a corresponding inequality for the numbers of double points, singularities, and inflections of closed contractible curves in the real projective plane which intersect every closed geodesic. In the process we will also obtain some generalizations of classical theorems due to Mobiu, Fenchel, and Segre (including Arnold’s “tennis ball theorem”). | ||
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| 17:40-18:20 | Qingchun Ji (Fudan University, P. R. China) | |
| “Division theorems for exact sequences” | ||
| Abstract: I will talk about Skoda-type division theorems for exact sequences of holomorphic vector bundles, and give applications to the Koszul complex. I wil also discuss how to use Skoda triples to establish global division theorems. | ||
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| Dec. 5 (Mon) | Morning Session | |
| 9:50-10:50 | Zizhou Tang (Beijing Normal University, P. R. China) | |
| “Gromov-Lawson-Schoen-Yau theory and isoparametric hypersurfaces” | ||
| Abstract: Motivated by the Gromov-Lawson-Schoen-Yau surgery theory on metrics of positive scalar curvature, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double manifold carries a metric of positive scalar curvature and an isoparametric foliation as well. | ||
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| 11:00-12:00 | Mu-Tao Wang (Columbia University, USA) | |
| “Mean curvature flows and isotopy problems” | ||
| Abstract: I shall discuss how mean curvature flows give canonical deformation of maps between Riemannian manifolds. Applications include estimations of null-homotopy constants of maps between spheres and smooth retractions of symplectomorphism groups of closed Riemann surfaces and complex projective spaces. | ||
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| Dec. 5 (Mon) | Afternoon Session | |
| 13:30-14:20 | Fuminori Nakata (Tokyo University of Science, Japan) | |
| “Integral transforms and the twistor theory for indefinite metrics” | ||
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Abstract:
Twistor theory for indefinite metrics, originated with LeBrun and Mason,
is progressing steadily. By this theory, one can establish one-to-one correspondence
between certain indefinite geometries and families of holomorphic disks on
complex manifolds. While the general theory for this type of twistor correspondence is studied, several explicit examples are constructed. These examples are described by making a use of Radon type integral transforms, and give a new insight to the theory of hyperbolic PDE’s. In this talk, an introduction to the LeBrun-Mason type twistor theory is given with showing examples and applications to hyperbolic PDE’s. |
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| 14:30-15:20 | Shouhei Honda (Kyushu University, Japan) | |
| “Convergence of Lipschitz functions and a weak second differentiable structure on limit spaces” | ||
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Abstract:
In this talk, we will give a new notion for convergence of Lipschitz
functions with respect to the Gromov-Hausdorff topology and several properties of
the convergence. As an application, we will show that all limit spaces of Riemannian
manifolds with lower Ricci curvature bounds have second differentiable structure in
some weak sense. Rererences: [1] S. Honda, Ricci curvature and convergence of Lipschitz functions, Commun. Anal. Geom. 19 (2011), 79-158. [2] S. Honda, A weak second differentiable structure on rectifiable metric measure spaces, preprint. |
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