Uniqueness of stable capillary hypersurfaces in a ball

In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. This solves completely a long-standing open problem. In the proof one of crucial ingredients is a new Minkowski type formula. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.Comment: Final version, Math. Ann., to appea

Classification of solutions of a Toda system in R^2

We consider solutions of a Toda system for SU(N+1) and show that any solution with finite exponential integral cam be obtained from a rational curve in complex projective space of dimension

Chern's magic form and the Gauss-Bonnet-Chern mass

In this note, we use Chern's magic form $\Phi_k$ in his famous proof of the Gauss-Bonnet theorem to define a mass for asymptotically flat manifolds. It turns out that the new defined mass is equivalent to the one that we introduced recently by using the Gauss-Bonnet-Chern curvature $L_k$. Moreover, this equivalence implies a simple proof of the equivalence between the ADM mass and the intrinsically defined mass via the Ricci tensor, which was reconsidered by Miao-Tam \cite{MT} and Herzlich \cite{H} very recently.Comment: 11 page

Geometric inequalities on locally conformally flat manifolds

Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.Comment: 30 pages. Final version, accepted by Duke Math.

An optimal anisotropic Poincar\'e inequality for convex domains

In this paper, we prove a sharp lower bound of the first (nonzero) eigenvalue of Finsler-Laplacian with the Neumann boundary condition. Equivalently, we prove an optimal anisotropic Poincar\'e inequality for convex domains, which generalizes the result of Payne-Weinberger. A lower bound of the first (nonzero) eigenvalue of Finsler-Laplacian with the Dirichlet boundary condition is also proved.Comment: 18 page

A new conformal invariant on 3-dimensional manifolds

By improving the analysis developed in the study of \s_k-Yamabe problem, we prove in this paper that the De Lellis-Topping inequality is true on 3-dimensional Riemannian manifolds of nonnegative scalar curvature. More precisely, if $(M^3, g)$ is a 3-dimensional closed Riemannian manifold with non-negative scalar curvature, then $$\int_M |Ric-\frac{\bar R} 3 g|^2 dv (g)\le 9\int_M |Ric-\frac{R} 3 g|^2dv(g),$$ where $\bar R=vol (g)^{-1} \int_M R dv(g)$ is the average of the scalar curvature $R$ of $g$. Equality holds if and only if $(M^3,g)$ is a space form. We in fact study the following new conformal invariant \ds \widetilde Y([g_0]):=\sup_{g\in {\cal C}_1([g_0])}\frac {\ds vol(g)\int_M \s_2(g) dv(g)} {\ds (\int_M \s_1(g) dv(g))^2}, where ${\cal C}_1([g_0]):=\{g=e^{-2u}g_0\,|\, R>0\}$ and prove that $\widetilde Y([g_0])\le 1/3$, which implies the above inequality.Comment: 23 page

Equivariant and Bott-type Seiberg-Witten Floer Homology: Part I

We construct Bott-type and equivariant Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. This paper is a revised version of math.DG/9701010. Some typos are removed.Comment: AMS Tex, 49 page

A conformal integral invariant on Riemannian foliations

Let $M$ be a closed manifold which admits a foliation structure $\mathcal{F}$ of codimension $q\geq 2$ and a bundle-like metric $g_0$. Let $[g_0]_B$ be the space of bundle-like metrics which differ from $g_0$ only along the horizontal directions by a multiple of a positive basic function. Assume $Y$ is a transverse conformal vector field and the mean curvature of the leaves of $(M,\mathcal{F},g_0)$ vanishes. We show that the integral $\int_MY(R^T_{g^T})d\mu_g$ is independent of the choice of $g\in [g_0]_B$, where $g^T$ is the transverse metric induced by $g$ and $R^T$ is the transverse scalar curvature. Moreover if $q\geq 3$, we have $\int_MY(R^T_{g^T})d\mu_g=0$ for any $g\in [g_0]_B$. However there exist codimension 2 minimal Riemannian foliations $(M,\mathcal{F},g)$ and transverse conformal vector fields $Y$ such that $\int_MY(R^T_{g^T})d\mu_g\neq 0$. Therefore, it is a nontrivial obstruction for the transverse Yamabe problem on minimal Riemannian foliation of codimension 2.Comment: 10 page

Analytic aspects of the Toda system: I. A Moser-Trudinger inequality

We analyze solutions of the Toda system and establish an optimal Moser-Trudinger inequalityComment: 35 page

On the Stability of Riemannian Manifold with Parallel Spinors

Inspired by the recent work of Physicists Hertog-Horowitz-Maeda, we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admits nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. In fact, we show that the Lichnerowicz Laplacian, which governs the second variation, is the square of a twisted Dirac operator. Our second result, which is a local version of the first one, shows that any metrics of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor. We also prove a rigidity result for special holonomy metrics. In the case of $SU(m)$ holonomy, the rigidity result implies that scalar flat deformations of Calabi-Yau metric must be Calabi-Yau. Finally we explore the connection with positive mass theorem, which presents another approach to proving these stability and rigidity results.Comment: revised version. references adde