Recent and Upcoming Talks
- Colloquium, Penn State University, September 2016
- Conference on Geometric Inequalities, Busan, South Korea November 2016
- AMS Sectional Meeting, Charleston, South Carolina, March 2017
- Geometry of Low Dimensional Manifolds, RIMS, Kyoto University, Japan, January 2018
Slides of Previous Talks
- Topology of Riemannian submanifolds with prescribed boundary
We prove that a smooth compact immersed submanifold of codimension 2 in R^{n}, n>2, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimenion is too high, or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem, and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman.
- Relative isoperimetric inequality outside convex bodies
We prove that the area of a hypersurface which traps a given volume outside of a convex body in Euclidean n-space must be greater than or equal to the area of a hemisphere trapping the given volume on one side of a hyperplane. The proof is based on a sharp estimate for total positive curvature of surfaces whose boundary meets a convex body orthogonally from the outside.
- Riemannian four vertex theorems
We prove that every metric of constant curvature on a compact surface M with boundary bdM induces at least four vertices, i.e., local extrema of geodesic curvature, on a connected component of bdM, if, and only if, M is simply connected. Indeed, when M is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant curvature on M which induce only two critical points of geodesic curvature on each component of bdM. With few exceptions, these metrics are obtained by removing the singularities and a perturbation of flat structures on closed surfaces. Further, we uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. We also characterize the simply connected space forms as the only surfaces in which every closed curve bounding a compact immersed surface has more than two vertices.
- Deformations of unbounded convex bodies and hypersurfaces
We study the topology of the space bd K^{n} of complete convex hypersurfaces of R^{n} which are homeomorphic to R^{n-1}. In particular, using Minkowski sums, we construct a deformation retraction of bd K^{n} onto the Grassmannian space of hyperplanes. So every hypersurface in bd K^{n} may be flattened in a canonical way. Further, the total curvature of each hypersurface evolves continuously and monotonically under this deformation. We also show that, modulo proper rotations, the subspaces of bd K^{n} consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg.
- Tangent cones and regularity of real hypersurfaces
We characterize C^{1} embedded hypersurfaces of R^{n} as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most m<3/2. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is C^{1}. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface X in R^{n} is C^{1}. Furthermore, if X is real algebraic, strictly convex, and unbounded, then its projective closure is a C^{1} hypersurface as well, which shows that X is the graph of a function defined over an entire hyperplane.
- Affine unfoldings of convex polyhedra
A well-known problem in geometry, which may be traced back to the Renaissance artist Albrecht Durer, is concerned with cutting a convex polyhedral surface along some spanning tree of its edges so that it may be isometrically embedded into the plane. We show that this is always possible after an affine transformation of the surface. In particular, unfoldability of a convex polyhedron does not depend on its combinatorial structure, which settles a problem of Croft, Falconer, and Guy. Among other techniques, the proof employs a topological characterization for embeddings among immersed planar disks.