434 research outputs found
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 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 . 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 is a 3-dimensional closed Riemannian manifold with
non-negative scalar curvature, then where is the average of the scalar curvature of . Equality holds if
and only if 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 and prove
that , 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 be a closed manifold which admits a foliation structure
of codimension and a bundle-like metric . Let be the
space of bundle-like metrics which differ from only along the horizontal
directions by a multiple of a positive basic function. Assume is a
transverse conformal vector field and the mean curvature of the leaves of
vanishes. We show that the integral
is independent of the choice of ,
where is the transverse metric induced by and is the transverse
scalar curvature. Moreover if , we have
for any . However there exist codimension 2 minimal Riemannian
foliations and transverse conformal vector fields such
that . 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
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
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