arXiv subjects

Jiabin Yin

Publications and source records attributed to Jiabin Yin.

17 recordsLinked to original sources

arXiv2026

Classification of compact Lagrangian self-similar submanifolds with Legendrian capillary boundary in the unit ball

We classify smooth compact connected Lagrangian immersions $X$ in the closed unit ball of $\C^n$, $n\ge2$, satisfying $H+\varepsilon X^\perp=0$, $\varepsilon\in\{-1,0,1\}$, with Legendrian boundary on the unit sphere and constant contact angle on each connected component. We prove that the boundary has at most two connected components. When the boundary is connected, $X$ is a diffeomorphism onto an equatorial Lagrangian $n$-disk. When the boundary has two components, $X$ splits globally as $X(s,p)=\gamma(s)\psi(p)$, where $\psi$ is a compact minimal Legendrian immersion in the unit sphere and $\gamma$ is an Anciaux profile with a unique radial minimum. The two contact angles are supplementary. In complex dimension two, every non-disk solution is a finite cover of a Lagrangian catenoid segment for $\varepsilon=0$ or of a rotational Anciaux annulus for $\varepsilon=\pm1$. In higher complex dimensions, iterated Calabi suspensions produce families whose minimal Legendrian links have nontrivial topology.

math.DG↗

arXiv2026

Estimates of $p$-capacity for manifolds with Ricci curvature bounded from below

We study sharp estimates for the $p$-capacity on complete non-compact Riemannian manifolds under lower Ricci curvature bounds. First, we establish sharp comparison inequalities for the $p$-capacity of bounded smooth domains in manifolds satisfying $\operatorname{Ric}\ge -ng.$ The estimates are expressed in terms of the boundary mean curvature and correspond to natural warped-product model ends. We characterize all equality cases and show that equality forces the exterior region to be isometric to the corresponding warped product. We also obtain an analogous sharp estimate under nonnegative Ricci curvature, whose equality case is described by an asymptotically flat model end. Second, we investigate normalized lower bounds for the relative $p$-capacity of condensers. We introduce scale-invariant quantities involving the volume of the inner set and the diameter of the ambient domain, establish uniform positive lower bounds, and determine the optimal ranges of the normalization parameters.

math.DG↗

arXiv2026

Mass-$p$-Capacity Inequalities in Asymptotically Flat Half-Spaces

In this paper, we establish general monotone quantities and sharp mass-capacity inequalities related to $p$-capacitary functions in $3$-dimensional asymptotically flat half-spaces of simple topology with nonnegative scalar curvature and nonnegative boundary mean curvature. These inequalities attain equality on a Schwarzschild half-space outside a rotationally symmetric half sphere.

math.DG↗

arXiv2026

Large $p$-Capacitary Invariants, Entropy, and Geometric Rank

We introduce large-$p$ asymptotic invariants associated with the $p$-capacity, the first $p$-eigenvalue, and the Maz'ya constant on connected complete noncompact Riemannian manifolds. For the two standard normalizations \[ p\,\operatorname{Cap}_p(\Omega)^{1/p} \quad\text{and}\quad (p-1)\operatorname{Cap}_p(\Omega)^{1/(p-1)}, \] we prove that their upper and lower limits are independent of the bounded smooth conductor $\Omega$. We denote the conductor-independent upper limit of the first normalization by $\mathcal C(M)$. When the second normalization converges to a positive limit, its logarithmic second-order coefficient is also conductor-independent. These invariants satisfy \[ \mathcal V(M)\geq \mathcal C(M)\geq \Lambda(M)=\mathcal M(M)\geq0. \] Under centered-ball isoperimetry or rotational symmetry, together with an eventual monotonicity assumption on the sphere-area-to-ball-volume ratio, all four invariants coincide with the volume entropy. This yields hyperbolic rigidity from either maximal $p$-spectral data or maximal $\mathcal C(M)$, as well as an almost-rigidity theorem under Ricci curvature and diameter bounds. For the universal cover of a closed negatively curved manifold, both capacitary normalizations converge to the volume entropy, which equals the topological entropy of the geodesic flow, without any centered-ball isoperimetric or rotational-symmetry assumption. For nonflat Hadamard manifolds with compact quotient satisfying either of the above geometric conditions, we obtain a second-order large-$p$ expansion whose logarithmic coefficient detects the geometric rank. Finally, sharp examples show that the general inequalities may be strict and that the first-order capacitary limit need not exist.

math.DG↗

arXiv2025

The first nonzero eigenvalue of the weighted p-Laplacian on differential forms

We introduce the weighted p-Laplace operator acting on differential forms on a metric measure space, which is a natural generalization of the p-Laplace operator defined by Seto [32]. We obtain some sharp lower bounds of the first nonzero eigenvalue for the weighted p-Laplacian. Our results extend an estimate of Seto [32], as well as the eigenvalue estimates derived by Cui-Sun [8] for closed submanifolds.

math.DG↗

arXiv2025

General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications

In this paper, we deal with an overdetermined problem for the $k$-Hessian equation ($1\leq k<\frac n2$) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve the problem, we give a new perspective to solve exterior overdetermined problem by combining two integral identities and geometric inequalities inspired by Brandolini-Nitsch-Salani's results \cite{BNS}. Meanwhile, we establish general monotone formulas to derive geometric inequalities related to $k$-admissible solution $u$ in $\mathbb R^n\setminusΩ$, where $Ω$ is smooth, $k$-convex and star-shaped domain, which constructed by Ma-Zhang\cite{MZ} and Xiao\cite{xiao}.

math.AP↗

arXiv2025

Stability and rigidity results of space-like hypersurface in the Minkowski space

In this paper, we establish some rigidity theorems for space-like hypersurfaces in Minkowski space by using a Weinberger-type approach with P-functions and integral identities. Firstly, for space-like hypersurfaces $M$ represented as graphs $x_{n+1}=u(x)$ over domain $\Omega\subset\mathbb R^n$, if higher-order mean curvature ratio $\frac{H_{k}}{H_l}(l<k)$ is constant and the boundary $\partial M$ lies on a hyperplane intersecting with constant angles, then the hypersurface must be a part of hyperboloid. Secondly, for convex space-like hypersurfaces with boundaries on a hyperboloid or light cone, if higher-order mean curvature ratio $\frac{H_{k}}{H_l}(l<k)$ is constant and the angle function between the normal vectors of the hypersurface and the hyperboloid (or the lightcone) on the boundary is constant, then such hypersurfaces must be a part of hyperboloid. These results significantly extend Gao's previous work presented in \cite{Gao1,Gao2}. Furthermore, we derive two fundamental integral identities for constant mean curvature (CMC) graphical hypersurfaces $x_{n+1}=u(x)$, $x\in\Omega\subset\mathbb R^n$, and the boundary lies on a hyperplane. As some applications: we obtain complete equivalence conditions for hyperboloid identification through curvature properties. We also establish a geometric stability estimate demonstrating that the square norm of the trace-free second fundamental form $\bar h$ of $M$ is quantitatively controlled by geometric quantities of $\partial\Omega$, as expressed by the inequality: $$ ||\bar h||_{L^2(\Omega)}\leq C(n,K)||H_{\partial\Omega}-H_0||_{L^1(\partial\Omega)}^{1/2}. $$ Here, $H_{\partial\Omega}$ is the mean curvature of $\partial\Omega$, $H_0$ is some reference constant and $C$ is a constant. Finally, analogous estimates are established.

math.DG↗

arXiv2025

Uniqueness of the critical points of solutions to two kinds of semilinear elliptic equations in higher dimensional domains

In this paper, we provide an affirmative answer to the {\it conjecture A} for bounded simple rotationally symmetric domains $Ω\subset \mathbb{R}^n(n\geq 3)$ along $x_n$ axis. Precisely, we use a new simple argument to study the symmetry of positive stable solutions for two kinds of semilinear elliptic equations. To do this, when $f(\cdot,s)$ is convex with respect to $s$, we show that the positivity of the first eigenvalue of the corresponding linearized operator in somehow symmetric domains is a sufficient condition for the symmetry of $u$. Moreover, we prove the uniqueness of critical points of a positive stable solution to semilinear elliptic equation $-\triangle u=f(\cdot,u)$ with zero Dirichlet boundary condition for simple rotationally symmetric domains in $\mathbb{R}^n$ by continuity method and a variety of maximum principles.

math.AP↗

arXiv2024

Willmore-type inequality for closed hypersurfaces in complete manifolds with Ricci curvature bounded below

In this paper, we establish a Willmore-type inequality for closed hypersurfaces in a complete Riemannian manifold of dimension $n+1$ with ${\rm Ric}\geq-ng$. It extends the classic result of Argostianiani, Fogagnolo, and Mazzieri in [1] to the Riemannian manifold of negative curvature. As an application, we construct a Willmore-type inequality for closed hypersurfaces in hyperbolic space and obtain the characterization of geodesic sphere.

math.DG↗

arXiv2023

New monotonicity for $p$-capacitary functions in $3$-manifolds with nonnegative scalar curvature

In this paper, we derive general monotone quantities and geometric inequalities associated with $p$-capacitary functions in asymptotically flat $3$-manifolds with simple topology and nonnegative scalar curvature. The inequalities become equalities on the spatial Schwarzschild manifolds outside rotationally symmetric spheres. This generalizes Miao's result \cite{M} from $p=2$ to $p\in (1, 3)$. As applications, we recover mass-to-$p$-capacity and $p$-capacity-to-area inequalities due to Bray-Miao \cite{BM} and Xiao \cite{Xiao}.

math.DG↗

arXiv2023

A partially overdetermined problem for $p$-Laplace equation in convex cones

We consider a partially overdetermined problem for the $p$-Laplace equation in a convex cone $\mathcal{C}$ intersected with the exterior of a smooth bounded domain $\overlineΩ$ in $\mathbb{R}^n$($n\geq2$). First, we establish the existence, regularity, and asymptotic behavior of a capacitary potential. Then, based on these properties of the potential, we use a $P$-function, the isoperimetric inequality, and the Heintze-Karcher type inequality in a convex cone to obtain a rigidity result under the assumption of orthogonal intersection.

math.AP↗

arXiv2021

On energy gap phenomena of the Whitney sphere and related problems

In this paper, we study Lagrangian submanifolds satisfying ${\rm \nabla^*} T=0$ introduced by Zhang \cite{Zh} in the complex space forms $N(4c)(c=0\ or \ 1)$, where $T ={\rm \nabla^*}\tilde{h}$ and $\tilde{h}$ is the Lagrangian trace-free second fundamental form. We obtain some Simons' type integral inequalities and rigidity theorems for such Lagrangian submanifolds. Moreover we study Lagrangian submanifolds in $\mathbb{C}^n$ satisfying $\nabla^*\nabla^*T=0$ and introduce a flow method related to them.

math.DG↗

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↗

arXiv2021

On $n$-dimensional complete self-similar solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ with nonnegative constant scalar curvature

As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo \cite{Guo} proved that $n$-dimensional compact self-shrinkers in $\mathbb{R}^{n+1}$ with scalar curvature bounded from above or below by some constant are isometric to the round sphere $\mathbb{S}^n(\sqrt{n})$, which implies that $n$-dimensional compact self-shrinkers in $\mathbb{R}^{n+1}$ with constant scalar curvature are isometric to the round sphere $\mathbb{S}^n(\sqrt{n})$(see also \cite{Hui1}). Complete classifications of $n$-dimensional translating solitons in $\mathbb{R}^{n+1}$ with nonnegative constant scalar curvature and of $n$-dimensional self-expanders in $\mathbb{R}^{n+1}$ with nonnegative constant scalar curvature were given by Martín, Savas-Halilaj and Smoczyk\cite{MSS} and Ancari and Cheng\cite{AC}, respectively. In this paper we give complete classifications of $n$-dimensional complete self-shrinkers in $\mathbb{R}^{n+1}$ with nonnegative constant scalar curvature. We will also give alternative proofs of the classification theorems due to Martín, Savas-Halilaj and Smoczyk \cite{MSS} and Ancari and Cheng\cite{AC}.

math.DG↗

arXiv2019

A new characterization of the Calabi torus in the unit sphere

In this paper, we study the rigidity theorem of closed minimally immersed Legendrian submanifolds in the unit sphere. Utilizing the maximum principle, we obtain a new characterization of the Calabi torus in the unit sphere which is the minimal Calabi product Legendrian immersion of a point and the totally geodesic Legendrian sphere. We also establish an optimal Simons' type integral inequality in terms of the second fundamental form of three dimensional closed minimal Legendrian submanifolds in the unit sphere.

math.DG↗

arXiv2019

Rigidity theorems of Lagrangian submanifolds in the homogeneous nearly Kähler $\mathbb{S}^6(1)$

In this paper, we study Lagrangian submanifolds of the homogeneous nearly Kähler $6$-dimensional unit sphere $\mathbb{S}^6(1)$. As the main result, we derive a Simons' type integral inequality in terms of the second fundamental form for compact Lagrangian submanifolds of $\mathbb{S}^6(1)$. Moreover, we show that the equality sign occurs if and only if the Lagrangian submanifold is either the totally geodesic $\mathbb{S}^3(1)$ or the Dillen-Verstraelen-Vrancken's Berger sphere $S^3$ discribed in J Math Soc Japan, 42: 565-584, 1990.

math.DG↗

arXiv2017

Equivariant CR minimal immersions from $S^3$ into $\mathbb{C}P^n$

The equivariant CR minimal immersions from the round $3$-sphere $S^3$ into the complex projective space $\mathbb CP^n$ have been classified by the third author explicitly (J London Math Soc 68: 223-240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant, and that all geometric properties of $S^3={\rm SU}(2)$ endowed with a left-invariant metric can be expressed in terms of the structure constants of the Lie algebra $\mathfrak{su}(2)$, we establish an extended classification theorem for equivariant CR minimal immersions from the $3$-sphere $S^3$ into $\mathbb CP^n$ without the assumption of constant sectional curvatures.

math.DG↗