Toru Sasahara
We investigate real hypersurfaces in complex space forms attaining equality in an inequality involving the contact $δ$-invariant $δ^c(2)$ introduced by Chen and Mihai in [3].
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We determine non-Hopf hypersurfaces with constant mean curvature in the complex projective plane which attain equality in a basic inequality between the maximum Ricci curvature and the squared mean curvature.
We provide a parametric construction in terms of minimal surfaces of the Euclidean submanifolds of codimension two and arbitrary dimension that attain equality in an inequality due to De Smet, Dillen, Verstraelen and Vrancken. The latter involves the scalar curvature, the norm of the normal curvature tensor and the length of the mean curvature vector.
Recently Oprea gave an improved version of Chen's inequality for Lagrangian submanifolds of $\mathbb CP^n(4)$. For minimal submanifolds this inequality coincides with the original previously proved version. We consider here those non minimal 3-dimensional Lagrangian submanifolds in $\mathbb CP^3 (4)$ attaining at all points equality in the improved Chen inequality. We show how all such submanifolds may be obtained starting from a minimal Lagrangian surface in $\mathbb CP^2(4)$ and a suitable horizontal curve in $S^3(1)$.
It was proved in Chen's paper \cite{chen} that every real hypersurface in the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$ δ(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is the mean curvature and $δ(2)$ is a $δ$-invariant introduced by him. In this paper, we study non-Hopf real hypersurfaces satisfying the equality case of the inequality under the condition that the mean curvature is constant along each integral curve of the Reeb vector field. We describe how to obtain all such hypersurfaces.
We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequality.
We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main theorem states that if the equality holds, then the hypersurface is a Schwarzschild solution. As part of our proof, we show that asymptotically flat graphical hypersurfaces with a minimal boundary and non-negative scalar curvature must be mean convex, using the argument that we deve…
Anisotropic extrinsic radius pinching for hypersurfaces and the stability of the Wulff shape
本文证明了涉及各向异性平均曲率的Hasanis--Koutroufiotis型外半径不等式，并研究了等号情形，证明几乎极值超曲面在Hausdorff距离意义下接近Wulff形状。
In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in $\mathbb{R}^{n+1}$ involving the norm of the covariant derivatives of both the difference tensor $K$ and the Tchebychev vector field $T$. Our result is optimal in that, applying our recent classification for locally strongly convex centroaffine hypersurfaces with parallel cubic form in [4], we can completely classify the hypersurfaces which realize the equality case of the inequality.
Rigidity of spacelike hypersurface with constant curvature and intersection angle condition
研究类空超曲面在Minkowski空间中的刚性性质，证明当k阶均曲率常数且边界与超平面交角恒定时，超曲面必为双曲面的一部分或完全包含于超平面中。
Some rigidity results on compact hypersurfaces with capillary boundary in Hyperbolic space
本文证明了双曲空间中毛细超曲面的Heintze-Karcher不等式，仅在完全脐曲超曲面时成立，并应用该结果证明了嵌入毛细超曲面的Alexandrov型定理，还证明了在双曲空间中支撑于全测地平面的毛细超曲面的其他刚性结果。
Heintze-Karcher inequality for anisotropic free boundary hypersurfaces in convex domains
该研究针对一般凸域中的各向异性自由边界超曲面，证明了最优Heintze-Karcher型不等式，明确等号成立条件，并借此推导出多种Alexandrov型定理，拓展了相关几何不等式研究。
In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces $Σ$ with nonnegative sectional curvature in $\mathbb{H}^n$. As an application, we prove the hyperbolic Alexandrov-Fenchel inequalities for hypersurfaces with nonnegative sectional curvature in $\mathbb{H}^n$: \begin{align*} \int_Σ p_{2k}\geq ω_{n-1}\left[\left(\frac{|Σ|}{ω_{n-1}}\right)^\frac{1}{k}+\left(\frac{|Σ|}{ω_{n-1}}\right)^{\frac{1}{k}\frac{n-1-2k}{n-1}}\right]^k, \end{align*} where $p_i$ is the normalized $i$-th mean curvature. Equality holds if and only if $Σ$ is…
Willmore-type inequalities for closed hypersurfaces in weighted manifolds
该研究针对具非负Bakry-Émery里奇曲率的加权流形闭超曲面，推导稳态与收缩梯度Ricci孤立子中的精确Willmore型不等式并刻画等号情形，得到相关经典不等式的加权版本，还推出加权等周型不等式。
The equality case in the substatic Heintze-Karcher inequality
本文证明亚静态流形中Heintze-Karcher不等式取等号时的刚性结论，并借此在扭曲乘积中完全去除Brendle常平均曲率超曲面刻画里的假设(H4)。
Motivated by the rigidity case in the localized Riemannian Penrose inequality, we show that suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality is necessarily smooth in properly specified coordinates. If applied to hypersurfaces enclosing the horizon in a spatial Schwarzschild manifold, the result gives the rigidity of isometric hypersurfaces with the same mean curvature.
Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the normal scalar curvature. This property is conformal invariant; hence we study them in the framework of Moebius geometry, and restrict to three dimensional Wintgen ideal submanifolds in S^5. In particular we give Moebius characterizations for minimal ones among them, which are also known as (3-dimensional) austere submanifolds (in 5-dimensional space forms).
On weakly Einstein submanifolds in space forms satisfying certain equalities
本文分类了空间形式中满足陈氏等式或半对称条件的弱爱因斯坦子流形与超曲面，并刻画了法联络平坦的情形。
Smooth approximation, Alexandrov-Fenchel type inequalities and curvature flows for h-convex bodies in hyperbolic space
该研究针对双曲空间中的h-凸体，通过球面卷积实现其光滑逼近，推广了两类Alexandrov-Fenchel不等式，引入曲率流证明其等号刚性，还建立了相关刚性定理以刻画测地球。
A submanifold in space forms satisfies the well-known DDVV inequality due to De Smet, Dillen, Verstraelen and Vrancken. The submanifold attaining equality in the DDVV inequality at every point is called Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are studied in this paper using the framework of Möbius geometry. We classify Wintgen ideal submanfiolds of dimension $m>2$ and arbitrary codimension when a canonically defined 2-dimensional distribution $\mathbb{D}$ is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super…
In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial Ω$ in $M$, with equality holding true if and only if $(M{\setminus}Ω, g)$ is isometric to a truncated cone over $\partialΩ$. An optimal version of Huisken's Isoperimetric Inequality for $3$-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version …
This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries are outermost minimal hypersurfaces. Specifically, we prove that if the Penrose Inequality is sufficiently close to being an equality on one of these manifolds, then it must be close to a Schwarzschild space with an appended cylinder, in the sense of Lipschitz Distance. Since the …
Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. Using the framework of Moebius geometry, we show that in the codimension two case, the mean curvature sphere of the Wintgen ideal submanifold corresponds to an 1-isotropic holomorphic curve in a complex quadric Q. Conversely, any 1-isotropic complex curve in Q describes a 2-parameter family of m-dimensional spheres whose envelope is always a m-dimensional Wintgen ideal submanifold at the re…
Suppose that $Σ=\partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $( M,\langle\;,\;\rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $Σ$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric $\langle\;,\;\rangle_H=H^2\langle\;,\;\rangle$ is at least $n/2$ and equality holds if and only if there exists a parallel spinor field on $ M$. As a consequence, if $Σ$ admits an isometric and isospin immersion $ϕ$ with mean curvature $H_0$ as a hypersurfac…
In this paper, based on the classical K. Yano's formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms, which involves the Ricci curvature in the direction $J\vec{H}$ and the norm of the covariant differentiation of the second fundamental form $h$, where $J$ is the almost complex structure and $\vec{H}$ is the mean curvature vector field. Second and analogously, for compact Legendrian submanifolds in the Sasakian space forms with Sasakian structure $(φ,ξ,η,g)$, we also establish an optimal integral inequality involving the Ricc…
Building on previous works of H. L. Bray, of P. Miao, and of S. Almaraz, E. Barbosa, and L. L. de Lima, we develop a doubling procedure for asymptotically flat half-spaces $(M,g)$ with horizon boundary $Σ\subset M$ and mass $m\in\mathbb{R}$. If $3\leq \dim(M)\leq 7$, $(M,g)$ has non-negative scalar curvature, and the boundary $\partial M$ is mean-convex, we obtain the Riemannian Penrose-type inequality $$ m\geq\left(\frac{1}{2}\right)^{\frac{n}{n-1}}\,\left(\frac{|Σ|}{ω_{n-1}}\right)^{\frac{n-2}{n-1}} $$ as a corollary. Moreover, in the case where $\partial M$ is not totally geodesic, we show …
We prove a sharp Alexandrov-Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic $n$-space, $n\geq 3$. The argument uses two new monotone quantities along the inverse mean curvature flow. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter-Schwarzschild solution. This sharpens previous results by Dahl-Gicquaud-Sakovich and settles, for this class of initial data sets, the conjectured Penrose i…
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