In the author's previous joint work with Hans-Joachim Hein, a mass formula for asymptotically locally Euclidean (ALE) Kaehler manifolds was proved, assuming only relatively weak fall-off conditions on the metric. However, the case of real dimension four presented technical difficulties that led us to require fall-off conditions in this special dimension that are stronger than the Chrusciel fall-off conditions that sufficed in higher dimensions. In this talk, I will explain how a new proof of the 4-dimensional case, using ideas from symplectic geometry, shows that Chrusciel fall-off suffices to imply all our main results in any dimension. In particular, I will explain why our Penrose-type inequality for the mass of an asymptotically Euclidean Kaehler manifold always still holds, given only this very weak metric fall-off hypothesis.
Isoparametric hypersurfaces in the sphere are those whose
principal curvatures are everywhere constant with fixed multiplicities. In
some sense, such hypersurfaces represent the simplest type of manifolds we
can get a handle on. They have rather complicated topology and most of them
are inhomogeneous, and thus they serve as a good testing ground for
constructing examples and counterexamples. The classification of such
hypersurfaces was initiated by E. Cartan around 1938, and the completion of
the last case with four principal curvatures will appear soon in
publication. Since the classification is a long story covering a wide
spectrum of mathematics, I will highlight in this talk the decisive moments
and the key ideas engaged in the intriguing pursuit.
The spectrum of the Laplace-Beltrami operator is
one of the fundamental invariants of a Riemannian manifold.
It has many applications, perhaps the most significant is in relation to
minimal surfaces. In the present talk we will show how minimal surfaces
arise in the study of isoperimetric inequalities for Laplace eigenvalues,
the relation that was initially discovered by P. Li and S. T. Yau. We will
present recent results in this direction and discuss connections to other
fields, including algebraic geometry and cobordism theory. The talk is based
on joint works with V. Medvedev, N. Nadirashvili, A. Penskoi and I.
Since the mid 1990’s, the leading candidate for a unified theory of all fundamental physical interactions has been M Theory.
A full formulation of M Theory is still not available, and it is only understood through its limits in certain regimes, which are either one of five 10-dimensional string theories, or 11-dimensional supergravity. The equations for these theories are mathematically interesting in themselves, as they reflect, either directly or indirectly, the presence of supersymmetry. We discuss recent progresses and open problems about two of these theories, namely supersymmetric compactifications of the heterotic string and of 11-dimensional supergravity. This is based on joint work of the speaker with Sebastien Picard and Xiangwen Zhang, and with Teng Fei and Bin Guo.
We generalize an inequality of E. Heintze and H. Karcher for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for k-Ricci curvature. Even in the case of a pointwise bound this generalizes the classical inequality by replacing a sectional curvature bound with a k-Ricci bound. This work is motivated by the estimates of Petersen-Shteingold-Wei for the volume of tubes around a geodesic and generalizes their estimate. Using similar ideas we also prove a Hessian comparison theorem for k-Ricci curvature which generalizes the usual Hessian and Laplacian comparison for distance functions from a point and give several applications.
A recent breakthrough of Wu and Yau asserts that a compact projective Kahler
manifold with negative holomorphic sectional curvature must have ample
canonical line bundle. In the talk, we will talk about some of the recent
advances along this direction. In particular, we will discuss the case
where the manifold is a noncompact Kahler manifold. We will also discuss
the case when the Kahlerity is a priori unknown. Part of these are joint
work with S. Huang, L.-F. Tam, F. Tong.
In this talk we will discuss the mean curvature flow with surgery and how to extend it to the low entropy, mean convex setting. An application to the topology of low entropy self shrinkers will also be discussed. This is a joint work with Shengwen Wang.
I will discuss a new construction of families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding singular fibration from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds. This is joint work with Hans-Joachim Hein, Song Sun, and Ruobing Zhang.