Given a Riemannian manifold M and a hypersurface H in M, it is well known that infinitesimal convexity on a neighborhood of a point in H implies local convexity. We show in this note that the same result holds in a semi-Riemannian manifold. We make some remarks for the case when only timelike, null or spacelike geodesics are involved. The notion of geometric convexity is also reviewed and some applications to geodesic connectedness of an open subset of a Lorentzian manifold are given.
In the realm of conformal geometry, we give a classification of the Euclidean hypersurfaces that admit a non-trivial conformal infinitesimal variation. In the restricted case of conformal variations, such a classification was obtained by E. Cartan in 1917. The case of infinitesimal isometric variations was done by U. Sbrana in 1908. In particular, we show that the class of hypersurfaces that allow a conformal infinitesimal variation is much larger than the one considered by Cartan.
We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the $k$-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve $k$-convexity in a Riemannia…
We prove that any $n$-dimensional closed mean convex $λ$-hypersurface is convex if $λ\le 0.$ This generalizes Guang's work on $2$-dimensional strictly mean convex $λ$-hypersurfaces. As a corollary, we obtain a gap theorem for closed $λ$-hypersurfaces with $λ\le 0.$
We establish the following Hadamard--Stoker type theorem: Let $f:M^n\rightarrow\mathscr{H}^n\times\mathbb R$ be a complete connected hypersurface with positive definite second fundamental form, where $\mathscr H^n$ is a Hadamard manifold. If the height function of $f$ has a critical point, then it is an embedding and $M$ is homeomorphic to $\mathbb S^n$ or $\mathbb R^n.$ Furthermore, $f(M)$ bounds a convex set in $\mathscr{H}^n\times\mathbb R.$ In addition, it is shown that, except for the assumption on convexity, this result is valid for hypersurfaces in $\mathbb S^n\times\mathbb R$ as well. …
Local Convexity and Planarity Estimates for Mean Curvature Flow
School of Mathematics；Statistics \& Key Laboratory of Nonlinear Analysis；Applications (Ministry of Education), Central China Normal University, Wuhan, 430079, P.R.China
It is known that the $L^{2}$-norms of a harmonic function over spheres satisfies some convexity inequality strongly linked to the Almgren's frequency function. We examine the $L^{2}$-norms of harmonic functions over a wide class of evolving hypersurfaces. More precisely, we consider compact level sets of smooth regular functions and obtain a differential inequality for the $L^{2}$-norms of harmonic functions over these hypersurfaces. To illustrate our result, we consider ellipses with constant eccentricity and growing tori in $\mathbf{R}^3.$ Moreover, we give a new proof of the convexity resul…
We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such hypersurfaces in hyperbolic space.
在[4]、[5]中，许多作者利用欧氏凸超曲面的模型和[1]中的思想，定义并研究了黎曼流形的凸超曲面。正如第一作者在[7]中所证明的，超曲面的凸性与高斯曲率之间存在相互依赖关系。本文定义了任意余维黎曼子流形的$H$ -凸性，用中曲率向量$H$代替超曲面的法向量. B. Y. Chen [2]，[3]作为严格$H$ -凸性的定义，得到了它.
We generalize a result of Giroux which says that a closed surface in a contact $3$-manifold with Morse-Smale characteristic foliation is convex. Specifically, we show that the result holds in contact manifolds of arbitrary dimension. As an application, we show that a particular closed hypersurface introduced by A. Mori is $C^{\infty}$-close to a convex hypersurface.
Given a smooth convex cone in the Euclidean $(n+1)$-space ($n\geq2$), we consider strictly mean convex hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve by a class of inverse curvature flows, then, by using the convexity of the cone in the derivation of the gradient and Hölder estimates, we can prove that this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of a round sphere as time tends to infinity.
In recent work, the notion of Double Convexity for a foliation of a conical null hypersurface was introduced to give a proof, if satisfied, of the Null Penrose Inequality. Double Convexity constrains the geometry of a Marginally Outer Trapped Surface (MOTS), called a quasi-round MOTS. In the first part of this paper, for a class of strictly stable Weakly Isolated Horizons, we show the existence of a unique foliation by quasi-round MOTS. In the second part, we show that any subsequent space-time perturbation continues to admit a quasi-round MOTS. Finally, for perturbations of the quasi-round MO…
We prove a sharp lower bound for the Tanaka-Webster holomorphic sectional curvature of strictly pseudoconvex real hypersurfaces that are "semi-isometrically" immersed in a Kähler manifold of nonnegative holomorphic sectional curvature under an appropriate convexity condition. This gives a partial answer to a question posed by Chanillo, Chiu, and Yang regarding the positivity of the Tanaka-Webster scalar curvature of the boundary of a strictly convex domain in $\mathbb{C}^2$ from 2012. In fact, the main result proves a stronger positivity property, namely the $\frac12$-positivity in the sense o…
