arXiv subjects

Chao Xia

Publications and source records attributed to Chao Xia.

At least 55 records · Page 3Linked to original sources

arXiv2022

Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball

In this paper, we first introduce the quermassintegrals for convex hypersurfaces with capillary boundary in the unit Euclidean ball $\mathbb{B}^{n+1}$ and derive its first variational formula. Then by using a locally constrained nonlinear curvature flow, which preserves the $n$-th quermassintegral and non-decreases the $k$-th quermassintegral, we obtain the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in $\mathbb{B}^{n+1}$. This generalizes the result in \cite{SWX} for convex hypersurfaces with free boundary in $\mathbb{B}^{n+1}$.

math.DG↗

arXiv2022

A partially overdetermined problem in domains with partial umbilical boundary in space forms

In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in \cite{GX} into the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze-Karcher-Ros type inequality for embedded hypersurfaces with free boundary lying on a horosphere or an equidistant hypersurface in the hyperbolic space. As an application, we show Alexandrov type theorem for constant mean curvature hypersurfaces with free boundary in these settings.

math.AP↗

arXiv2022

A Heintze-Karcher type inequality for hypersurfaces with capillary boundary

In this paper, we establish a Heintze-Karcher type inequality for hypersurfaces with capillary boundary of contact angle $θ\in (0,\fracπ{2})$ in a half space or a half ball, by using solution to a mixed boundary value problem in Reilly type formula. Consequently, we give a new proof of Alexandrov type theorem for embedded capillary constant mean curvature hypersurfaces with contact angle $θ\in (0,\fracπ{2})$ in a half space or a half ball.

math.DG↗

arXiv2021

Stable capillary hypersurfaces supported on a horosphere in the hyperbolic space

In this paper, we study a stability problem of free boundary hypersurfaces, and also capillary ones whose boundary supported on a horosphere in hyperbolic space. We prove that umbilical hypersurfaces are only stable immersed capillary hypersurfaces whose boundary supported on a horosphere. Using the same method, we show that a totally geodesic hyperplane is only stable immersed type-II hypersurface whose boundary supported on a horosphere.

math.DG↗

arXiv2021

Symmetrization with respect to mixed volumes

In this paper we introduce new symmetrization with respect to mixed volume or anisotropic curvature integral, which generalizes the one with respect to quermassintegral due to Talenti and Tso. We show a Pólya-Szego type principle for such symmetrization -- it diminishes the anisotropic Hessian integral for quasi-convex functions. We achieve this by a systematic study of invariants on non-symmetric matrices with real eigenvalues and the higher order anisotropic mean curvatures of level sets, which may be of independent interest. As applications, we establish a comparison principle for anisotropic Hessian equations and sharp anisotropic Sobolev inequalities.

math.AP↗

arXiv2021

Anisotropic $p$-capacity and anisotropic Minkowski inequality

In this paper, we prove a sharp anisotropic $L^p$ Minkowski inequality involving the total $L^p$ anisotropic mean curvature and the anisotropic $p$-capacity, for any bounded domains with smooth boundary in $\mathbb{R}^n$. As consequences, we obtain an anisotropic Willmore inequality, a sharp anisotropic Minkowski inequality for outward $F$-minimising sets and a sharp volumetric anisotropic Minkowski inequality. For the proof, we utilize a nonlinear potential theoretic approach which has been recently developed in \cite{AFM1}.

math.AP↗

arXiv2020

Two-Stage Copy-Move Forgery Detection with Self Deep Matching and Proposal SuperGlue

Copy-move forgery detection identifies a tampered image by detecting pasted and source regions in the same image. In this paper, we propose a novel two-stage framework specially for copy-move forgery detection. The first stage is a backbone self deep matching network, and the second stage is named as Proposal SuperGlue. In the first stage, atrous convolution and skip matching are incorporated to enrich spatial information and leverage hierarchical features. Spatial attention is built on self-correlation to reinforce the ability to find appearance similar regions. In the second stage, Proposal SuperGlue is proposed to remove false-alarmed regions and remedy incomplete regions. Specifically, a proposal selection strategy is designed to enclose highly suspected regions based on proposal generation and backbone score maps. Then, pairwise matching is conducted among candidate proposals by deep learning based keypoint extraction and matching, i.e., SuperPoint and SuperGlue. Integrated score map generation and refinement methods are designed to integrate results of both stages and obtain optimized results. Our two-stage framework unifies end-to-end deep matching and keypoint matching by obtaining highly suspected proposals, and opens a new gate for deep learning research in copy-move forgery detection. Experiments on publicly available datasets demonstrate the effectiveness of our two-stage framework.

cs.CV↗

arXiv2020

Guan-Li type mean curvature flow for free boundary hypersurfaces in a ball

In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.

math.DG↗

arXiv2020

Stability for a second type partitioning problem

In this paper, we study stability and instability problem for type-II partitioning problem. First, we make a complete classification of stable type-II stationary hypersurfaces in a ball in a space form as totally geodesic $n$-balls. Second, for general ambient spaces and convex domains, we give some topological restriction for type-II stable stationary immersed surfaces in two dimension. Third, we give a lower bound for the Morse index for type-II stationary hypersurfaces in terms of their topology.

math.DG↗

arXiv2019

What Will Your Child Look Like? DNA-Net: Age and Gender Aware Kin Face Synthesizer

Visual kinship recognition aims to identify blood relatives from facial images. Its practical application-- like in law-enforcement, video surveillance, automatic family album management, and more-- has motivated many researchers to put forth effort on the topic as of recent. In this paper, we focus on a new view of visual kinship technology: kin-based face generation. Specifically, we propose a two-stage kin-face generation model to predict the appearance of a child given a pair of parents. The first stage includes a deep generative adversarial autoencoder conditioned on ages and genders to map between facial appearance and high-level features. The second stage is our proposed DNA-Net, which serves as a transformation between the deep and genetic features based on a random selection process to fuse genes of a parent pair to form the genes of a child. We demonstrate the effectiveness of the proposed method quantitatively and qualitatively: quantitatively, pre-trained models and human subjects perform kinship verification on the generated images of children; qualitatively, we show photo-realistic face images of children that closely resemble the given pair of parents. In the end, experiments validate that the proposed model synthesizes convincing kin-faces using both subjective and objective standards.

cs.LG↗

arXiv2019

A partially overdetermined problem in a half ball

In this paper, we study a partially overdetermined mixed boundary value problem in a half ball. We prove that a domain in which this partially overdetermined problem admits a solution if and only if the domain is a spherical cap intersecting $ß^{n-1}$ orthogonally. As an application, we show a stationary point for a partially torsional rigidity under a volume constraint must be a spherical cap.

math.AP↗

arXiv2019

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.

math.DG↗

arXiv2018

Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkowski-type inequality and for $n=3$ we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.

math.DG↗

arXiv2018

Motion of level sets by inverse anisotropic mean curvature

In this paper we consider the weak formulation of the inverse anisotropic mean curvature flow, in the spirit of Huisken-Ilmanen. By using approximation method involving Finsler-p-Laplacian, we prove the existence and uniqueness of weak solutions.

math.AP↗

arXiv2017

Locally constrained inverse curvature flows

We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order, namely the radial coordinate and the generalized support function. Under various assumptions we prove longtime existence and smooth convergence to a coordinate slice. We apply this result to deduce a new Minkowski type inequality in the anti-de-Sitter Schwarzschild manifolds and a weighted isoperimetric type inequality in the hyperbolic space.

math.DG↗

arXiv2017

$L^p$ Christoffel-Minkowski problem: the case $1< p<k+1$

We consider a fully nonlinear partial differential equation associated to the intermediate $L^p$ Christoffel-Minkowski problem in the case $1<p<k+1$. We establish the existence of convex body with prescribed $k$-th even $p$-area measure on $\mathbb S^n$, under an appropriate assumption on the prescribed function. We construct examples to indicate certain geometric condition on the prescribed function is needed for the existence of smooth strictly convex body. We also obtain $C^{1,1}$ regularity estimates for admissible solutions of the equation when $ p\ge \frac{k+1}2$.

math.DG↗

arXiv2017

Inverse anisotropic mean curvature flow and a Minkowski type inequality

In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the $C^\infty$ topology. As an application, we prove a Minkowski type inequality for star-shaped, $F$-mean convex hypersurfaces.

math.DG↗

arXiv2017

An integral formula for affine connections

In this article, we introduce a $2$-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in \cite{LX} for substatic manifolds. On the other hand, this technique leads to various geometric inequalities and eigenvalue estimates under a much more general Ricci curvature conditions. The new Ricci curvature condition interpolates between static Ricci tensor and $1$-Bakry-Emery Ricci, and also includes the conformal Ricci as an intermediate case.

math.DG↗