Hypersurfaces In Riemannian Manifolds Research Articles

Comparison formulas for total mean curvatures of Riemannian hypersurfaces

Abstract We devise some differential forms after Chern to compute a family of formulas for comparing total mean curvatures of nested hypersurfaces in Riemannian manifolds. This yields a quicker proof of a recent result of the author with Joel Spruck, which had been obtained via Reilly’s identities.

Stability of isometric immersions of hypersurfaces

Abstract We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$ -perturbations of their fundamental forms: For a manifold ${\mathcal M}^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:{\mathcal M}^d\to {\mathcal N}^{d+1}$ , whose pullback metrics and shape operators are arbitrary close in $L^p$ to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold ${\mathcal N}$ , removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.

Density of a thin film billiard reflection pseudogroup in a Hamiltonian symplectomorphism pseudogroup

Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary C∞-smooth hypersurface γ ⊂ ℝn+1 that is either a global strictly convex closed hypersurface, or a germ of a hypersurface. We deal with the pseudogroup generated by compositional ratios of reflections from γ and of reflections from its small deformations. In the case when γ is a global convex hypersurface, we show that the C∞-closure of the latter pseudogroup contains the pseudogroup of Hamiltonian diffeomorphisms between domains in the phase cylinder: the space of oriented lines intersecting γ transversally. We prove an analogous local result in the case when γ is a germ. The derivatives of the above compositional differences in the deformation parameter are Hamiltonian vector fields calculated by Ron Perline. To prove the main results, we find the Lie algebra generated by them and prove its C∞-density in the Lie algebra of Hamiltonian vector fields. We also prove analogues of the above results for hypersurfaces in Riemannian manifolds.

Заметка о -квазиомбилических гиперплоскостях почти эрмитовых многообразий

In the present note, we consider the introduced by Lidia Vasil’evna Stepanova notion of an -quasi-umbilical hypersurface in an almost Her­mitian manifold. We show that the notion of an -quasi-umbilical hyper­surface in an almost Hermitian manifold is connected with the notion of a minimal hypersurface in this manifold. Using the classical theory of minimal hypersurfaces in Riemannian manifolds and Kirichenko — Stepanova general theory of almost contact metric hypersurfaces in almost Hermitian manifolds, we establish that an -quasi-umbilical hypersurface of a nearly Kählerian manifold is minimal if and only if this hypersurface is totally umbilical. Taking into account the connection between the notions of a minimal hypersurface and of an -quasi-umbilical hypersurface in an almost Her­mitian manifold, we conclude that some well-known results in the theory of almost contact metric hypersurfaces in almost Hermitian manifolds can be reformulated. The problem of the existence of a non-umbilical minimal -quasi-umbilical hypersurface of a quasi-Kählerian manifold is posed.

Min-max theory for free boundary minimal hypersurfaces, I: Regularity theory

In the 1960s, Almgren [3, 4] initiated a program to find minimal hypersurfaces in Riemannian manifolds using min-max method. This program was largely advanced by Pitts [34] and Schoen–Simon [37] in the 1980s when the manifold is compact without boundary. In this paper, we finish this program for general compact manifold with non-empty boundary. As a corollary, we establish the existence of a smooth embedded minimal hypersurface with non-empty free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves [31] shows that for any compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary, there exist infinitely many properly embedded minimal hypersurfaces with non-empty free boundary.

Mean curvature flow in Fuchsian manifolds

Motivated by the goal of detecting minimal surfaces in hyperbolic manifolds, we study geometric flows in complete hyperbolic [Formula: see text]-manifolds. In general, the flows might develop singularities at some finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic [Formula: see text]-manifolds (Fuchsian manifolds) which are warped products of a closed surface of genus at least two and [Formula: see text]. We show that for a large class of closed initial surfaces, which are graphs over the totally geodesic surface [Formula: see text], the mean curvature flow exists for all time and converges to [Formula: see text]. This is among the first examples of converging mean curvature flows starting from closed hypersurfaces in Riemannian manifolds. We also provide calculations for the general warped product setting which will be useful for further works.

Inverse curvature flow for hypersurfaces in Riemannian manifold and its application

This is a survey paper on the inverse curvature flow for hypersurfaces in Riemannian manifold.We first discuss the long time behavior of the inverse curvature flow in Euclidean space, and its application in proving the Alexandrov-Fenchel inequalities for star-shaped hypersurfaces.Then we discuss the related results in hyperbolic space and in sphere.Finally, we discuss the inverse mean curvature flow in Kottler space. Kottler space is an important example of warped product space,and is aymptotically locally hyperbolic at the infinity and satisfies the static equation. We will consider the convergence result of inverse mean curvature flow in such space and also discussits application in proving the Minkowski-type inequality for star-shaped and mean convex hypersurfaces. Inverse curvature flow is an active research area in recent years.We cannot include all results in this short article. For the convenience of the interested readers, we list a few related references on other topics that we do not mention.

Compactness analysis for free boundary minimal hypersurfaces

We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and derive various consequences in terms of finiteness and topological control.

A fully nonlinear flow for two-convex hypersurfaces in Riemannian manifolds

We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity $$G_\kappa = \big ( \sum _{i < j} \frac{1}{\lambda _i+\lambda _j-2\kappa } \big )^{-1}$$ , where $$\lambda _1 \le \cdots \le \lambda _n$$ denote the curvature eigenvalues and $$\kappa $$ is a nonnegative constant. This defines a fully nonlinear parabolic equation, provided that $$\lambda _1+\lambda _2>2\kappa $$ . In contrast to mean curvature flow, this flow preserves the condition $$\lambda _1+\lambda _2>2\kappa $$ in a general ambient manifold. Our main goal in this paper is to extend the surgery algorithm of Huisken–Sinestrari to this fully nonlinear flow. This is the first construction of this kind for a fully nonlinear flow. As a corollary, we show that a compact Riemannian manifold satisfying $$\overline{R}_{1313}+\overline{R}_{2323} \ge -2\kappa ^2$$ with non-empty boundary satisfying $$\lambda _1+\lambda _2 > 2\kappa $$ is diffeomorphic to a 1-handlebody. The main technical advance is the pointwise curvature derivative estimate. The proof of this estimate requires a new argument, as the existing techniques for mean curvature flow due to Huisken–Sinestrari, Haslhofer–Kleiner, and Brian White cannot be generalized to the fully nonlinear setting. To establish this estimate, we employ an induction-on-scales argument; this relies on a combination of several ingredients, including the almost convexity estimate, the inscribed radius estimate, as well as a regularity result for radial graphs. We expect that this technique will be useful in other situations as well.

Poincaré and mean value inequalities for hypersurfaces in Riemannian manifolds and applications

In the first part of this paper we prove some new Poincar\'e inequalities, with explicit constants, for domains of any hypersurface of a Riemannian manifold with sectional curvatures bounded from above. This inequalities involve the first and the second symmetric functions of the eigenvalues of the second fundamental form of such hypersurface. We apply these inequalities to derive some isoperimetric inequalities and to estimate the volume of domains enclosed by compact self-shrinkers in terms of its scalar curvature. In the second part of the paper we prove some mean value inequalities and as consequences we derive some monotonicity results involving the integral of the mean curvature.

Curvature estimates for immersed hypersurfaces in Riemannian manifolds

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of $(\mathbb S^2, g)$ in $3$-dimensional warped product space $(N^3, \bar g)$. We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter-Schwarzschild manifolds and the Reissner-Nordstr\"om manifolds.

A filtration for isoparametric hypersurfaces in Riemannian manifolds

Motivated by the well known Cartan-Munzner equations and the Chern conjecture, we introduce the notion of k-isoparametric hypersurface in an (n + 1)- dimensional Riemannian manifold for k = 0,1,...,n. Many fundamental and inter- esting results ( towards the classification of homogeneous hypersurfaces among other things ) are given in complex projective spaces, complex hyperbolic spaces, and even in locally rank one symmetric spaces. Further investigations in this direction are expected.

The ADM mass of asymptotically flat hypersurfaces

We provide integral formulae for the ADM mass of asymptotically flat hypersurfaces in Riemannian manifolds with a certain warped product structure in a neighborhood of infinity, thus extending Lam's recent results on Euclidean graphs to this broader context. As applications we exhibit, in any dimension, new classes of manifolds for which versions of the Positive Mass and Riemannian Penrose inequalities hold and discuss a notion of quasi-local mass in this setting. The proof explores a novel connection between the co-vector defining the ADM mass of a hypersurface as above and the Newton tensor associated to its shape operator, which takes place in the presence of an ambient Killing field.

Kinematics of Hypersurfaces in Riemannian Manifolds

In this work we study the kinematics of an m-dimensional continuum moving in an (m+1)-dimensional ambient space, modelled by the motion of a hypersurface M in a Riemannian manifold N. Focusing on the codimension of the continuum relative to the ambient space rather than on its dimension, we provide a unified framework for either curves moving on surfaces, surfaces moving in a 3-dimensional space (Euclidean or Riemannian), or hypersurfaces moving in a Riemannian space of arbitrary dimension. Further, the use of general geometric structures and a coordinate-free language facilitate the description of more general continua. We present formulae for the variation of geometric quantities of the hypersurface, some of which generalize those given for surfaces in Euclidean space (Kadianakis, N. in J. Elasticity 16:1–17, 2010). The main point in the present work is the use of kinematical quantities which result from a generalized version of the polar decomposition theorem for surfaces introduced by Man, C.-S. and Cohen, H. (in J. Elast. 16:97–104, 1986) and adapted here for hypersurfaces. We apply our results to motion along the normal and to pure strain motion. Finally, we discuss the relation of our results to Differential Geometry and Physics. Specifically, we provide an application for the case of a curve moving on a 2-dimensional manifold, which models a 1-dimensional continuum moving on a surface. The later application is presented independently of the embedding of the surface in a larger space.

Vanishing theorems on hypersurfaces in Riemannian manifolds

We study a complete noncompact stable minimal hypersurface M and a strongly stable hypersurface M with constant mean curvature in a 5-dimensional Riemannian manifold N. If N is a compact simply connected manifold with bounded sectional curvature 517≤K̄≤1, then there is no nontrivial L2 harmonic form on M. This is a generalized version of Tanno’s result on a stable minimal hypersurface in R5 and Zhu’s result on a stable minimal hypersurface in S5.

Caccioppoli’s inequalities on constant mean curvature hypersurfaces in Riemannian manifolds

We prove some Caccioppoli’s inequalities for the traceless part of the second fundamental form of a complete, noncompact, finite index, constant mean curvature hypersurface of a Riemannian manifold, satisfying some curvature conditions. This allows us to unify and clarify many results scattered in the literature and to obtain some new results. For example, we prove that there is no stable, complete, noncompact hypersurface in $${{\mathbb R}^{n+1}, n \leq 5}$$ , with constant mean curvature $${H \not=0}$$ , provided that, for suitable p, the L p norm of the traceless part of second fundamental form satisfies some growth condition.

Biharmonic hypersurfaces in Riemannian manifolds

We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied in \cite{Ji2}, \cite{CH}, \cite{CMO1}, \cite{CMO2}. We then apply the equation to show that the generalized Chen's conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a (2-parameter) family of conformally flat metrics and a (4-parameter) family of multiply warped product metrics each of which turns the foliation of an upper-half space of $\mathhbb{R}^m$ by parallel hyperplanes into a foliation with each leave a proper biharmonic hypersurface. We also characterize proper biharmonic vertical cylinders in $S^2\times \mathbb{R}$ and $H^2\times \mathbb{R}$.

Constant k-curvature hypersurfaces in Riemannian manifolds

In [7], Rugang Ye (1991) proved the existence of a family of constant mean curvature hypersurfaces in an ( m + 1 ) -dimensional Riemannian manifold ( M m + 1 , g ) , which concentrate at a point p 0 (which is required to be a nondegenerate critical point of the scalar curvature), moreover he proved that this family constitutes a foliation of a neighborhood of p 0 . In this paper we extend this result to the other curvatures (the r-th mean curvature for 1 ⩽ r ⩽ m ). And we give the expansion of the m-dimensional volume of the leaves of this foliation as well as the ( m + 1 ) -dimensional volume of the sets enclosed by each leaf.

An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds

It is proved that a convex hypersurface in a Riemannian manifold of sectional curvature $\ge\kappa$ is an Alexandrov's space of curvature $\ge\kappa$. This theorem provides an optimal lower curvature bound for an older theorem of Buyalo.

Curvature estimates for Weingarten hypersurfaces in Riemannian manifolds

We prove curvature estimates for general curvature functions. As an application we show the existence of closed, strictly convex hypersurfaces with prescribed curvature F, where the defining cone of F is Γ+. F is only assumed to be monotone, symmetric, homogeneous of degree 1, concave and of class Cm,α, m ≥ 4.