arXiv subjects

Liangjun Weng

Publications and source records attributed to Liangjun Weng.

At least 19 recordsLinked to original sources

arXiv2026

Neumann problem for Hessian equation and geometric inequalities in hyperbolic space

We establish the existence of solutions to the Neumann problem for the Hessian equation on a smooth, bounded strictly static convex domain in hyperbolic space. The boundary condition determines a unique compatibility constant, and admissible solutions are unique up to a multiple of the static potential. As an application, we give a direct analytic proof of some weighted geometric inequalities for the static convex domain in hyperbolic space.

math.AP↗

arXiv2026

Quermassintegral inequalities for capillary hypersurfaces in a horoball

In this paper, we study the quermassintegral inequalities for capillary hypersurfaces in a horoball in hyperbolic space. We construct a locally constrained nonlinear curvature flow that preserves the $k$-th quermassintegral while monotonically increasing the $l$-th quermassintegral for any $(1\leq l<k\leq n)$. Using this flow, we establish a full family of the quermassintegral inequalities for horo-convex capillary hypersurfaces in a horoball with contact angle $θ\in(0,π/2]$. Moreover, we weaken the horo-convexity to convexity and obtain the quermassintegral inequalities for convex capillary hypersurfaces between the $n$-th and $l$-th quermassintegrals.

math.DG↗

arXiv2026

Prescribed $L_{p}$ curvature problem for convex capillary hypersurface

We address the prescribed $L_p$ curvature problem for convex capillary hypersurfaces in the Euclidean half-space. Using the support function, we formulate the problem as a Hessian quotient equation on a spherical cap with a Robin boundary condition. For $p\geq1$, we establish existence results under suitable assumptions on the prescribed data, together with uniqueness in the corresponding parameter ranges. At the critical exponent, we obtain an associated eigenvalue problem with uniqueness up to dilation. In the subcritical range, existence is established for capillary even data. As applications, we obtain existence results for the capillary $L_p$ Christoffel--Minkowski problem and prescribed $L_p$ quotient curvature problems, providing a framework for studying these equations in the presence of a capillary boundary.

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arXiv2026

Uniqueness of capillary Gauss solitons

We prove the rigidity conjecture of [16, Conjecture 1.2] for smooth strictly convex capillary Gauss solitons in a Euclidean half-space with an acute contact angle: every such soliton is a spherical cap. Combined with our previous convergence result for the capillary Gauss curvature flow [16, Theorem 1.1], it follows that the flow starting from a strictly convex capillary hypersurface with an acute contact angle converges to a capillary spherical cap, after a suitable rescaling.

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arXiv2026

The capillary $L_p$-Minkowski problem

This paper is a continuation of our recent work [Adv. Math. 469 (2025), Paper No. 110230] concerning the capillary Minkowski problem. We propose, in this paper, a capillary $L_p$-Minkowski problem for $p\in \mathbb{R}$, which seeks to find a capillary convex body with a prescribed capillary $L_p$-surface area measure in the Euclidean half-space. This formulation provides a natural Robin boundary analogue of the classical $L_p$-Minkowski problem introduced by Lutwak [J. Differential Geom. 38 (1993), no. 1, 131--150]. For $p>1$, we resolve the capillary $L_p$-Minkowski problem in the smooth category by reducing it to a Monge--Ampère equation with a Robin boundary condition on the unit spherical cap.

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arXiv2025

The capillary Christoffel-Minkowski problem

In this article, we introduce a $k$-th capillary area measure for capillary convex bodies in the Euclidean half-space, which serves as a boundary counterpart to the classical concept of area measure (see, e.g., \cite[Chapter 8]{Sch}). We then propose a Christoffel-Minkowski problem for capillary convex bodies, to find a capillary convex body in the Euclidean half-space with a prescribed $k$-th capillary area measure. This problem is equivalent to solving a Hessian-type equation with a Robin boundary value condition. We then establish the existence and uniqueness of a smooth solution under a natural sufficient condition.

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arXiv2025

The capillary Gauss curvature flow

In this article, we first introduce a Gauss curvature type flow for capillary hypersurfaces, which we call capillary Gauss curvature flow. We then show that the flow will shrink to a point in finite time. This is a capillary counterpart (or Robin boundary counterpart) of Firey's problem studied in [Mathematika 21 (1974), pp. 1-11] and Tso [Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882]. Finally, we prove that its normalized flow converges to a soliton. This is a capillary counterpart of the result of Guan and Ni in [J. Eur. Math. Soc. 19 (2017), no. 12, 3735-3761]. The classification of solitons remains an open conjecture.

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arXiv2025

Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary II

In this paper, we provide an affirmative answer to [16, Conjecture 1.5] on the Alexandrov-Fenchel inequality for quermassintegrals for convex capillary hypersurfaces in the Euclidean half-space. More generally, we establish a theory for capillary convex bodies in the half-space and prove a general Alexandrov-Fenchel inequality for mixed volumes of capillary convex bodies. The conjecture [16, Conjecture 1.5] follows as its consequence.

math.MG↗

arXiv2025

Convex capillary hypersurfaces of prescribed curvature problem

In this paper, we study the prescribed $k$-th Weingarten curvature problem for convex capillary hypersurfaces in $\overline{\mathbb{R}^{n+1}_+}$. This problem naturally extends the prescribed $k$-th Weingarten curvature problem for closed convex hypersurfaces, previously investigated by Guan-Guan in [19], to the capillary setting. We reformulate the problem as the solvability of a Hessian quotient equation with a Robin boundary condition on a spherical cap. Under a natural sufficient condition, we establish the existence of a strictly convex capillary hypersurface with the prescribed $k$-th Weingarten curvature. This also extends our recent work on the capillary Minkowski problem in [40].

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arXiv2024

Prescribed $L_p$ quotient curvature problem and related eigenvalue problem

In this paper, we investigate the existence of admissible (and strictly convex) smooth solutions to the prescribed $L_p$ quotient type curvature problem with $p>1$. For cases where $p=k-l+1$ and $p> k-l+1$, we obtain an admissible solution without any additional conditions, which is strictly spherically convex under a convexity condition. Under the same convexity condition, we establish the existence of a strictly spherically convex solution for the case $p<k-l+1$, provided that the prescribed function is even, a condition known to be necessary.

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arXiv2024

A fully nonlinear locally constrained curvature flow for capillary hypersurface

In this article, we study a locally constrained fully nonlinear curvature flow for convex capillary hypersurfaces in half-space. We prove that the flow preserves the convexity, exists for all time, and converges smoothly to a spherical cap. This can be viewed as the fully nonlinear counterpart of the result in \cite{MWW}. As a byproduct, a high-order capillary isoperimetric ratio (1.6) evolves monotonically along this flow, which yields a class of the Alexandrov-Fenchel inequalities.

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arXiv2024

A Minkowski-type inequality for capillary hypersurfaces in a half-space

In this article, we investigate a flow of inverse mean curvature type for capillary hypersurfaces in the half-space. We establish the global existence of solutions for this flow and demonstrate that it converges smoothly to a spherical cap as time tends to infinity. As a result, we derive a new Minkowski-type inequality for star-shaped and mean convex capillary hypersurfaces for the whole range of contact angle $θ\in (0,π)$.

math.AP↗

arXiv2024

Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space

In this article, we first introduce the quermassintegrals for compact hypersurfaces with capillary boundaries in hyperbolic space from a variational viewpoint, and then we solve an isoperimetric type problem in hyperbolic space. By constructing a new locally constrained inverse curvature flow, we obtain the Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space. This generalizes a theorem of Brendle-Guan-Li \cite{BGL} for convex closed hypersurfaces in hyperbolic space.

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arXiv2024

Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow

In this paper, we introduce a volume- or area-preserving curvature flow for hypersurfaces with capillary boundary in the half-space, with speed given by a positive power of the mean curvature with a non-local averaging term. We demonstrate that for any convex initial hypersurface with a capillary boundary, the flow exists for all time and smoothly converges to a spherical cap as $t \to +\infty$

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arXiv2024

The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space

In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds with free boundary. We first generalize ideas of restricted normal cones introduced by Choe-Ghomi-Ritoré in \cite{CGR06} and obtain an optimal area estimate for generalized restricted normal cones. This area estimate, together with the ABP method of Cabré in \cite{Cabre2008}, provides a new proof of the relative isoperimetric inequality obtained by Choe-Ghomi-Ritoré in \cite{CGR07}. Furthermore, we use this estimate and the idea of Brendle in his recent work \cite{Brendle2019} to obtain a relative isoperimetric inequality for minimal submanifolds with free boundary on a convex support surface in $\mathbb{R}^{n+m}$, which is optimal and gives an affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open Problem 12.6, when the codimension $m\leq 2$.

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arXiv2023

A constrained mean curvature type flow for capillary boundary hypersurfaces in space forms

In this paper, we introduce a new constrained mean curvature type flow for capillary boundary hypersurfaces in space forms. We show the flow exists for all time and converges globally to a spherical cap. Moreover, the flow preserves the volume of the bounded domain enclosed by the hypersurface and decreases the total energy. As a by-product, we give a flow proof of the capillary isoperimetric inequality for the starshaped capillary boundary hypersurfaces in space forms.

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arXiv2023

A constrained mean curvature flow and Alexandrov-Fenchel inequalities

In this article, we study a locally constrained mean curvature flow for star-shaped hypersurfaces with capillary boundary in the half-space. We prove its long-time existence and the global convergence to a spherical cap. Furthermore, the capillary quermassintegrals defined in \cite{WWX2022} evolve monotonically along the flow, and hence we establish a class of new Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary in the half-space.

math.DG↗

arXiv2023

Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary

In this paper, we first introduce quermassintegrals for capillary hypersurfaces in the half-space. Then we solve the related isoperimetric type problems for the convex capillary hypersurfaces and obtain the corresponding Alexandrov-Fenchel inequalities. In order to prove these results, we construct a new locally constrained curvature flow and prove that the flow converges globally to a spherical cap.

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