The 10th Pacific RIM Geometry Conference 2011 Osaka-Fukuoka: Part II

Dec. 7 (Wed) Morning Session
9:40-9:50 Opening Remarks
9:50-10:50 Henry Wente (The University of Toledo, USA)
“Exotic capillary tubes”
Abstract: In contrast to the standard capillary tube, an exotic capillary tube is a rotationally symmetric tube of variable cross-section which if positioned correctly in a vessel of fluid possesses a continuum of equilibrium configurations. The controlling variables are the capillary constant k = ρg/σ and the contact angle γ. Lowering the tube slightly from its natural position causes the tube to completely fill up while raising the tube slightly forces the tube to drain out. Other surprising consequences follow. Potential commercial applications will also be discussed.
Note: My paper discussing this topic recently appeared in the “Journal of Mathematical Fluid Mechanics” 13 (2011) 355-370.
11:00-12:00 Jaigyoung Choe (KIAS, Korea)
“A sharp isoperimetric inequality for minimal surfaces”
Abstract: It is proved that a minimal surface with no genus and three boundary components in R3 satisfies the classical isoperimetric inequality 4πA ≤ L2. (Joint work with Richard Schoen)
Dec. 7 (Wed) Afternoon Session
13:30-14:30 Miyuki Koiso (Kyuhsu University/PREST, JST, Japan)
“Pitchfork bifurcation for hypersurfaces with constant mean curvature”
Abstract: We consider hypersurfaces with constant mean curvature with given boundary conditions. Choosing the mean curvature H or the volume V enclosed by the hypersurface as parameter, we construct conditions under which a pitchfork bifurcation occurs. We apply our results to isoperimetric problems in the Riemannian products of S1 and simply connected space forms introduced by Pedrosa and Ritoré (1999).
14:40-15:20 Keomkyo Seo (Sookmyung Women’s University, Korea)
“L2 harmonic 1-forms on a complete minimal submanifold in hyperbolic space”
Abstract: We study the nonexistence of L2 harmonic 1-forms and topological property on minimal submanifolds in hyperbolic space. We also estimate the first eigenvalue for the Laplacian operator on minimal submanifolds in hyperbolic space.
15:50-16:20 Rung-Tzung Huang (National Central University, Taiwan)
“The comparison theorem of the refined analytic torsions on manifolds with boundary for an acyclic Hermitian connection”
Abstract: The refined analytic torsion was introduced by M. Braverman and T. Kappeler as a canonical refinement of analytic torsion on odd dimensional closed Riemannian manifolds. It is defined by using the graded zeta-determinant of the odd signature operator. The refined analytic torsion on compact Riemannian manifolds with boundary has been discussed by B. Vertman and by Y. Lee and myself, but these two constructions are completely different. In this talk we will discuss the comparison theorem of these two constructions when the odd signature operator comes from an acyclic Hermitian flat connection.
16:20-17:00 Toshiaki Omori (Tohoku U., Japan)
“On existence of harmonic maps via exponentially harmonic maps”
Abstract: An exponenially harmonic map u : (M, g) → (N, h) is a map between compact Riemannian manifolds which extremize the functional
E(u) := ∫ M e |∇u|2 dvolg.
A remarkable fact on them is that, unlike harmonic maps, exponentially harmonic maps are known to always exist in a given homotopy class H ∈ [M,N] and to be necessarily smooth. In the present talk, I would like to introduce an approximation of a harmonic map via a sequence of exponentially harmonic maps. Also, if possible, I would like to mention a time-evolution equation for exponentially harmonic maps, which is also expected to well approximate harmonic maps.
17:00-17:30 Sung-Hong Min (KIAS, Korea)
“Embeddedness of proper minimal submanifolds in homogeneous spaces”
Abstract: In this presentation, we briefly show the three embeddedness results as follows.
(1) Let Γ2m+1 be a polygon with 2m + 1 vertices in Rn, where m is an integer ≥ 2. Then the total curvature of Γ2m+1 < 2mπ. In particular, the total curvature of Γ5 < 4π and thus any minimal surface Σ ⊂ Rn bounded by Γ5 is embedded. Let Γ5 be a piecewise geodesic Jordan curve with 5 vertices in Hn. Then any minimal surface Σ ⊂ Hn bounded by Γ5 is embedded. If Γ5 is in a geodesic ball of radius π/4 in Sn+, then Σ ⊂ Sn+ is also embedded. As a consequence, Γ5 is an unknot in R3, H3 and S3+.
(2) Let Σ be an m-dimensional proper minimal submanifold in Hn with the ideal boundary ∂Σ = Γ in the infinite sphere Sn−1 = ∂Hn. If the Möbius volume vol~(Γ) of Γ satisfies vol~(Γ) < 2vol(Sm−1), then Σ is embedded. If vol~(Γ) = 2vol(Sm−1), then Σ is embedded unless it is a cone.
(3) Let Σ be a proper minimal surface in H2 × R. If Σ is vertically regular at infinity and has two ends, then Σ is embedded.
Dec. 8 (Thu) Morning Session
9:50-10:50 Paolo Piccione (University of Sao Paulo, Brazil)
“Equivariant bifurcation in geometric variational problem”
Abstract: I will first discuss some abstract equivariant bifurcation results for variational problems. Then I will present some applications, including bifurcation of constant mean curvature embeddings, bifurcation of solutions of the Yamabe problems in product manifolds and in some special Riemannian submersions, and bifurcation of solutions of the σ2-Yamabe problem in product of Einstein manifolds.
11:00-12:00 Yoonweon Lee (Inha University, Korea)
“Gluing formula of the refined analytic torsion”
Abstract: The refined analytic torsion was introduced by Braverman and Kappeler on an odd dimensional closed Riemannian manifold in 2000’s as an analytic analogue of the Turaev torsion. It is defined by using the spectrum of the odd signature operator and is described as an element of the determinant line for cohomologies. Specially, when the odd signature operator is defined by an acyclic Hermitian connection, the refined analytic torsion is a complex number whose modulus part is a classical Ray-Singer analytic torsion and the phase part is the rho invariant, the difference of two eta invariants. In earlier work we introduced a well-posed boundary condition for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this talk we discuss the gluing formula for the refined analytic torsion on a closed Riemannian manifold with respect to this boundary condition in case that the odd signature operator is defined by an acyclic Hermitian connection. Basic tools are BFK-gluing formula for zeta-determinants and the gluing formula of eta invariant given by Brüning, Lesch and Kirk.
Dec. 8 (Thu) Afternoon Session
13:30-14:30 Entao Zhao (Zhejiang U., P.R.China)
“The mean curvature flow in higher codimensions”
Abstract: The mean curvature flow is the negative gradient flow of the volume functional of the submanifolds. Many convergence theorems have been proved for the mean curvature flow of hypersurfaces or of submanifolds with low dimension or admitting some special structure. Recently, B. Andrews and C. Baker proved some beautiful convergence theorems for the mean curvature flow of submanifolds in an Euclidean space or a sphere.
In this talk, I will discuss several new convergence theorems for mean curvature flow in higher codimensions. The initial submanifold is assumed to satisfy suitable pointwise or integral curvature pinching conditions. The talk is based on the joint works with Kefeng Liu, Hongwei Xu and Fei Ye.
14:40-15:20 Yu Kawakami (Yamaguchi U., Japan)
“A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space”
Abstract: We provide an effective ramification theorem for the ratio of canonical forms of weakly complete flat fronts in the hyperbolic three-space. As an application, we give a simple proof of the classification of complete nonsingular flat surfaces in the hyperbolic three-space.
Dec. 8 (Thu) Ph. D. Students Session
15:50-16:15 Renato Bettiol (University of Notre Dame, USA)
“Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres”
Abstract: We study local rigidity and bifurcation of the Yamabe problem on 1-parameter families of homogeneous metrics on spheres. More precisely, we use variational techniques to study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics. The abstract tools used are an implicit function theorem and a bifurcation criterion relying on jumps of the Morse index, which also have applications to other geometric variational problems. In the case of the Yamabe problem, this means that finding bifurcation instants amounts to computing the spectrum of the Laplacian and scalar curvature of each metric in the family. Applying this to 1-parameter families of U(n + 1), Sp(n + 1) and Spin(9)-homogeneous metrics, we prove local rigidity in the first and existence of infinitely many bifurcation instants in the last two families. As corollaries, we obtain some global uniqueness and multiplicity results on such families.
[Joint work with P. Piccione.]
16:15-16:40 Nguyen Thac Dung (National Tsinghua U., Taiwan)
“Complete Smooth Metric Measure Spaces with Spectrum Bounded from Below”
Abstract: Join work with Prof. Chiung Jue Sung (National Tsinghua Univeristy). We consider smooth metric measure spaces (M, g, e−fdv) with weighted Laplacian Δf. Assuming λ1f) is bounded from below in term of |grad f| and its Bakry-Émery curvature bounded from below by λ1f), we prove the rigidity of M. This result generalizes the work of Li and Wang (see [2]) on complete non-compact Riemannian manifolds and extends the work of Munteanu and Wang (see [4]) on the smooth metric measure spaces with its Bakry-Émery curvarute bounded from below. At the same time, we address the space of f-harmonic functions with finite f-energy. A structure property of this space is given if the Bakry-Émery bounded from below.
References:
[1] N. T. Dung and C. J. Sung, Some Remarks on Manifolds with positive spectrum, preprint.
[2] P. Li and J. Wang, Complete manifolds with positive spectrum, Jour. Diff. Geom. 58 (2001) 501-534.
[3] O. Munteanu and J. Wang, Smooth metrci measure spaces with non-negative curvature, Comm. Anal. Geom. 19 (2011), no. 3, 451-486.
[4] O. Munteanu and J. Wang, Analysis of weighted Laplacian and application to Ricci solitons, Preprint.
16:40-17:05 Abdullah Kizilay (Tohoku University, Japan)
“Viscosity solutions on a Riemannian manifold”
Abstract: In this talk, we consider viscosity solutions to second order partial differential equations, in particular Cauchy-Dirichlet problem [CDP] of the form; ut + F(t,x,u,Du,D2u) = 0 with a boundary condition and an initial condition at time t = 0. We work on Riemannian manifold and present the structures of semijets, comparison principle and Perron’s method. Existence, uniqueness and stability results for the Cauchy-Dirichlet problem on a Riemanian manifold are studied.
17:10-17:35 Kotaro Kawai (Tohoku University, Japan)
“Construction of special Lagrangian submanifolds”
Abstract: The notion of special Lagrangian submanifolds was introduced by Harvey and Lawson in 1982. They are important in mirror symmetry due to the SYZ conjecture. In this short talk, I will construct special Lagrangian submanifolds in non-flat spaces explicitly. From the construction, these spaces are fibered by special Lagrangian submanifolds.
17:35-18:00 Satoshi Ueki (Tohoku University, Japan)
“Leaf-wise intersections in coisotropic submanifolds”
Abstract: J. Moser considered the leaf-wise intersection which is motivated by perturbation theory of periodic orbits for Hamiltonian systems. The leaf-wise intersection is a generalization of the Lagrangian intersection and the fixed point of symplectomorphisms. In this talk, we consider the first existence theorem of leafwise intersections by Moser and the history of this existence problem.
Dec. 9 (Fri) Morning Session
9:50-10:50 Boris Botvinnik (University of Oregon, USA)
“Surgery, concordance and isotopy of metrics with positive scalar curvature”
Abstract: Two positive scalar curvature metrics g0, g1 on a manifold M are psc-isotopic if they are homotopic through metrics of positive scalar curvature. It is well known that if two metrics g0, g1 of positive scalar curvature on a closed compact manifold M are psc-isotopic, then they are psc-concordant, i.e. there exists a metric g of positive scalar curvature on the cylinder M×I which has zero mean curvature along the boundary and extends the metrics g0 on M×{0} and g1 on M×{1}. In my lecture, I will discuss the problem whether a psc-concordance implies psc-isotopy. There is a combination of relevant methods to be used here: surgery tools related to Gromov-Lawson construction, classic results on isotopy and pseudo-isotopy of diffeomorphisms, standard geometric analysis related to the conformal Laplacian, and the Ricci flow.
File of Talk
11:00-12:00 Harish Seshadri (Indian Institute of Science, Bangalore, India)
“On Wilking’s criterion for the Ricci fow”
Abstract: Recently Wilking gave a simple criterion for generating Ricci flow invariant nonnegative curvature conditions which recovers most of the known such conditions: In dimension n let S be an AdSO(n,C)-invariant subset of so(n,C). Then the cone C(S) of curvature curvature operators which are positive on S is invariant under Ricci flow.
In this talk we show that the class of AdSO(n,C)-invariant subset of so(n,C) is a disjoint union of two subclasses with the following properties:
(i) Let S belong to the first class. Then the connected sum of manifolds whose curvature operators lie in C(S) also admits a metric with curvature operator in C(S).
(ii) If S is in the second class then the normalized Ricci flow of a manifold whose curvature operator lies in C(S) converges to a metric of constant positive sectional curvature.
File of Talk
Dec. 9 (Fri) Afternoon Session
13:30-14:30 Kazuo Akutagawa (Tohoku University, Japan)
“3-manifolds with positive flat conformal structure”
Abstract: In this talk, we consider a closed 3-manifold M with flat conformal structure C. We will show that, if the Yamabe constant of (M,C) is positive, then (M,C) is Kleinian. This result is probably a folk theorem, by using Witten’s approach to positive mass theorems. Here, we will give another explicit proof of it.
File of Talk
14:40-15:30 Bennett Palmer (Idaho State University, USA)
“Anisotropic variational problems for surfaces”
Abstract: We will discuss recent progress in the study of surfaces which are critica of an anisotropic surface energy. This will include results for free boundary problems and the anisotropic mean curvature flow.
File of Talk
15:30-16:00 Naoya Ando (Kumamoto University, Japan)
“Hopf’s theorem for surfaces with constant mean curvature and its generalizations”
Abstract: It is well-known that if a surface with constant mean curvature in E3 is homeomorphic to a sphere, then the surface is a round sphere. The same conclusion holds in the case where the surface is special Weingarten. Koiso-Palmer obtained an analogous result for surfaces with constant anisotropic mean curvature. The purpose of this talk is to introduce the outline of a proof by the speaker of Koiso-Palmer’s theorem.
File of Talk