arXiv subjects
Publications and source records attributed to Haizhong Li.
We give a Green-function proof of the stable Bernstein theorem in $\mathbb R^7$ for smooth, connected, complete, two-sided minimal hypersurface, thus resolving the last case in stable Bernstein problem.
We study the spiral product $G_γ(t,x,y)=(z_1(t)f_1(x),z_2(t)f_2(y))$, which couples two spherical $\mathscr C$-totally real immersed factors through a profile curve $γ=(z_1,z_2)\subset\mathbb S^3$. Its minimality is governed by a weighted-geodesic system: $G_γ$ is minimal precisely when both factors are minimal and $γ$ is an unparametrized geodesic of $|z_1|^{2k_1}|z_2|^{2k_2}g_{\mathbb S^3}$. This flow is Liouville integrable and, when $k_1+k_2>0$, its phase map has an open dense full-rank locus. Consequently, closed profiles of arbitrarily large primitive order occur densely. For compact minimal factors, Routh reduction and Sturm oscillation give $\text{Ind}(G_γ)\geq \text{Ind}(f_1)+\text{Ind}(f_2)+2m_γ-3$ for a closed oscillatory profile of primitive closing order $m_γ$. On the contact level, a factor-adapted choice of profiles produces, from any prescribed pair of compact connected embedded special Legendrians, embedded special Legendrian products of every sufficiently large prime closing order. Finally, canonical finite horizontal lifts and Hopf projection give Delaunay-type minimal Lagrangians in complex projective spaces. For compact embedded inputs and ordinarily closed profiles, the primitive spherical quotient is embedded, while its projective quotient is embedded exactly when the reduced relative winding is one.
In this paper, we establish the Weinstock inequality for the first non-zero Steklov eigenvalue on star-shaped mean convex domains in hyperbolic space $\mathbb{H}^n$ for $n\geq 3$. We note that when $n\geq 4$, the result was obtained in our previous paper [23]. In particular, when the domain is convex, our result gives an affirmative answer to Open Question 4.27 in [13] for the hyperbolic case.
In this paper, we study the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$. This problem was recently posed by Sadovsky and Zhang. First, by a second variation argument and a dimension reduction construction, we show that the inequality fails for arbitrary convex bodies when $q>n$, and fails even in the origin-symmetric class when $q>n+2$. Secondly, we prove the endpoint case $q=n+2$ for origin-symmetric convex bodies via Hadwiger's inequality for the polar moment of inertia. Finally, for unconditional convex bodies, we establish the inequality in the full range $0<q\le n+1$ by using a singular weighted Reilly formula and a coordinate-slice Hardy inequality. As applications, we derive several uniqueness results for the corresponding dual curvature measures.
In this paper, we classify the hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$ with constant sectional curvature. In contrast to $\mathbb{S}^2\times\mathbb{S}^2$, the resulting examples for $\mathbb{H}^2\times\mathbb{H}^2$ exhibit more diversity, and we construct a special example with non-constant product angle function. For $\mathbb{S}^2\times\mathbb{S}^2$, however, the product angle function of any constant sectional curvature hypersurface is identically zero. As a byproduct, we classify the hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$ with constant product angle function and constant mean curvature (or constant scalar curvature).
In Euclidean space $\mathbb{R}^n$, the minimization problem of a nonlocal isoperimetric functional with a competition between perimeter and a nonlocal term derived from the negative power of the distance function, has been extensively studied. In this paper, we investigate this nonlocal isoperimetric problem in hyperbolic space $\mathbb{H}^n$, we prove that the geodesic balls are unique minimizers (up to hyperbolic isometries) for small volumes $m$ and obtain nonexistence results for large volumes $m$ under certain ranges of the exponent in the nonlocal term.
In this paper, we study the Hopf hypersurfaces of the complex hyperbolic quadric $Q^{m*}=SO^o_{2,m}/(SO_2\times SO_m)$ ($m\geq3$) with constant principal curvatures. We classify the Hopf hypersurfaces of $Q^{m*}$ ($m\geq3$) with at most two distinct constant principal curvatures. For Hopf hypersurfaces with three or four distinct constant principal curvatures, we determine the values of the principal curvatures as well as their multiplicities.
In this paper, we propose a new definition of stable $(r+1)$-th capillary hypersurfaces from variational perspective for any $1\leq r\leq n-1$. More precisely, we define stable $(r+1)$-th capillary hypersurfaces to be smooth local minimizers of a new energy functional under volume-preserving and contact angle-preserving variations. Using the new concept of the stable $(r+1)$-th capillary hypersurfaces, we generalize the stability results of Souam \cite{Souam} in a Euclidean half-space and Guo-Wang-Xia \cite{GWX} in a horoball in hyperbolic space for capillary hypersurface to $(r+1)$-th capillary hypersurface case.
In this paper, we classify the Hopf hypersurfaces of the complex quadric $Q^m=SO_{m+2}/(SO_2SO_m)$ ($m\geq3$) with at most five distinct constant principal curvatures. We also classify the Hopf hypersurfaces of $Q^m$ ($m=3,4,5$) with constant principal curvatures. All these real hypersurfaces are open parts of homogeneous examples.
In this paper, we prove that complete noncompact constant mean curvature hypersurfaces in $\mathbb{R}^6$ with finite index must be minimal. This provides a positive answer to do Carmo's question in dimension $6$. The proof strategy is also applicable to $\mathbb{R}^4$ and $\mathbb{R}^5$, thereby providing alternative proofs for those previously resolved cases.
In this paper, the notion of hyperbolic ellipsoids in hyperbolic space is introduced. Using a natural orthogonal projection from hyperbolic space to Euclidean space, we establish affine isoperimetric type inequalities for static convex domains in hyperbolic space. Moreover, equality of such inequalities is characterized by these hyperbolic ellipsoids.
In this paper, we prove Heintze-Karcher type inequalities involving the shifted mean curvature for smooth bounded domains in certain sub-static warped product manifolds. In particular, we prove a Heintze-Karcher-type inequality for non mean-convex domains in the hyperbolic space. As applications, we obtain uniqueness results for hypersurfaces satisfying a class of curvature equations.
The $L_p$-Christoffel-Minkowski problem and the prescribed $L_p$-Weingarten curvature problem for convex hypersurfaces in Euclidean space are important problems in geometric analysis. In this paper, we consider their counterparts in hyperbolic space. For the horospherical $p$-Christoffel-Minkowski problem first introduced and studied by the second and third authors, we prove the existence of smooth, origin-symmetric, strictly horospherically convex solutions by establishing a new full rank theorem. We also propose the prescribed $p$-shifted Weingarten curvature problem and prove an existence result.
This note aims to give a better understanding and some remarks about recent preprint ``Spiral Minimal Products". In particular, 1. it should be pointed out that a generalized Delaunay construction among minimal Lagrangians of complex projective spaces has been set up. This is a general structural result working for immersion and current situations. 2. uncountably many new regular (or irregular) special Lagrangian cones with finite density and ``regular" (or irregular) special Lagrangian cones with infinite density in complex Euclidean spaces can be found.
In this paper, we establish the Weinstock inequality for the first non-zero Steklov eigenvalue on star-shaped mean convex domains in hyperbolic space $\mathbb{H}^n$ for $n \geq 4$. In particular, when the domain is convex, our result gives an affirmative answer to Open Question 4.27 in [7] for the hyperbolic space $\mathbb{H}^n$ when $n \geq 4$.
In this paper, we extend several results established for stable minimal hypersurfaces to $δ$-stable minimal hypersurfaces. These include the regularity and compactness theorems for immersed $δ$-stable minimal hypersurfaces in $\mathbb{R}^{n+1}$ when $n \geq 3$ and $δ> \frac{n-2}{n}$, as well as the $δ$-stable Bernstein theorem for $n=3$ and $n=4$ for properly immersion. The range of $δ$ is optimal, as the $n$-dimensional catenoid in $\mathbb{R}^{n+1}$ is $\frac{n-2}{n}$-stable.
In \cite{LX}, the first author and Xu introduced and studied the horospherical $p$-Minkowski problem in hyperbolic space $\mathbb{H}^{n+1}$. In particular, they established the uniqueness result for solutions to this problem when the prescribed function is constant and $p\ge -n$. This paper focuses on the isotropic horospherical $p$-Minkowski problem in hyperbolic plane $\mathbb{H}^{2}$, which corresponds to the equation \begin{equation}\label{0} φ^{-p}\left(φ_{θθ}-\frac{φ_θ^2}{2φ}+\frac{φ-φ^{-1}}{2}\right)=γ\quad\text{on}\ \mathbb{S}^1, \end{equation} where $γ$ is a positive constant. We provide a classification of solutions to the above equation for $p\ge -7$, as well as a nonuniqueness result of solutions for $p<-7$. Furthermore, we extend this problem to the isotropic horospherical $q$-weighted $p$-Minkowski problem in hyperbolic plane and derive some uniqueness and nonuniqueness results.
This paper exhibits a structural strategy to produce new minimal submanifolds in spheres based on two given ones. The method is to spin the given minimal submanifolds by a curve $γ\subset \mathbb S^3$ in a balanced way and leads to resulting minimal submanifolds $-$ spiral minimal products, which form a two-dimensional family arising from intriguing pendulum phenomena decided by $C$ and $\tilde C$. With $C=0$, we generalize the construction of minimal tori in $\mathbb S^3$ explained in [Bre13] to higher dimensional situations. When $C=-1$, we recapture previous relative work in [CLU06] and [HK12] for special Legendrian submanifolds in spheres, and moreover, can gain numerous $\mathscr C$-totally real and totally real embedded minimal submanifolds in spheres and in complex projective spaces respectively. A key ingredient of the paper is to apply a beautiful extension result of minimal submanifolds by Harvey and Lawson [HL75] for a rotational reflection principle in our situation to establish curve $γ$.