Hypersurfaces In Manifolds Research Articles

Embeddedness of min–max CMC hypersurfaces in manifolds with positive Ricci curvature

We prove that on a compact Riemannian manifold of dimension 3 or higher, with positive Ricci curvature, the Allen–Cahn min–max scheme in Bellettini and Wickramasekera (The Inhomogeneous Allen–Cahn Equation and the Existence of Prescribed-Mean-Curvature Hypersurfaces, 2020), with prescribing function taken to be a non-zero constant λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}, produces an embedded hypersurface of constant mean curvature λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document} (λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-CMC). More precisely, we prove that the interface arising from said min–max contains no even-multiplicity minimal hypersurface and no quasi-embedded points (both of these occurrences are in principle possible in the conclusions of Bellettini and Wickramasekera, 2020). The immediate geometric corollary is the existence (in ambient manifolds as above) of embedded, closed λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-CMC hypersurfaces (with Morse index 1) for any prescribed non-zero constant λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}, with the expected singular set when the ambient dimension is 8 or higher.

Multiplicity‐1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature

AbstractWe address the one‐parameter minmax construction for the Allen–Cahn energy that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian manifold with (Guaraco's work, relying on works by Hutchinson, Tonegawa, and Wickramasekera when sending the Allen–Cahn parameter to 0). We obtain the following result: if the Ricci curvature of N is positive then the minmax Allen–Cahn solutions concentrate around a multiplicity‐1 minimal hypersurface (possibly having a singular set of dimension ). This multiplicity result is new for (for it is also implied by the recent work by Chodosh–Mantoulidis). We exploit directly the minmax characterization of the solutions and the analytic simplicity of semilinear (elliptic and parabolic) theory in . While geometric in flavour, our argument takes advantage of the flexibility afforded by the analytic Allen–Cahn framework, where hypersurfaces are replaced by diffused interfaces; more precisely, they are replaced by sufficiently regular functions (from N to ), whose weighted level sets give rise to diffused interfaces. We capitalise on the fact that (unlike a hypersurface) a function can be deformed both in the domain N (deforming the level sets) and in the target (varying the values). We induce different geometric effects on the diffused interface by using these two types of deformations; this enables us to implement in a continuous way certain operations, whose analogues on a hypersurface would be discontinuous. An immediate corollary of the multiplicity‐1 conclusion is that every compact Riemannian manifold with and positive Ricci curvature admits a two‐sided closed minimal hypersurface, possibly with a singular set of dimension at most . (This geometric corollary also follows from results obtained by different ideas in an Almgren–Pitts minmax framework.)

Area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below

Abstract In this paper, we study area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below with Cheeger–Colding theory. Let N i {N_{i}} be a sequence of smooth manifolds with Ricci curvature ≥ - n ⁢ κ 2 {\geq-n\kappa^{2}} on B 1 + κ ′ ⁢ ( p i ) {B_{1+\kappa^{\prime}}(p_{i})} for constants κ ≥ 0 {\kappa\geq 0} , κ ′ > 0 {\kappa^{\prime}>0} , and volume of B 1 ⁢ ( p i ) {B_{1}(p_{i})} has a positive uniformly lower bound. Assume B 1 ⁢ ( p i ) {B_{1}(p_{i})} converges to a metric ball B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} in the Gromov–Hausdorff sense. For a sequence of area-minimizing hypersurfaces M i {M_{i}} in B 1 ⁢ ( p i ) {B_{1}(p_{i})} with ∂ ⁡ M i ⊂ ∂ ⁡ B 1 ⁢ ( p i ) {\partial M_{i}\subset\partial B_{1}(p_{i})} , we prove the continuity for the volume function of area-minimizing hypersurfaces equipped with the induced Hausdorff topology. In particular, each limit M ∞ {M_{\infty}} of M i {M_{i}} is area-minimizing in B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} provided B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} is a smooth Riemannian manifold. By blowing up argument, we get sharp dimensional estimates for the singular set of M ∞ {M_{\infty}} in ℛ {\mathcal{R}} , and 𝒮 ∩ M ∞ {\mathcal{S}\cap M_{\infty}} . Here, ℛ {\mathcal{R}} and 𝒮 {\mathcal{S}} are the regular and singular parts of B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} , respectively.

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.

Minimal hypersurfaces in manifolds of Ricci curvature bounded below

Abstract In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding’s method in [T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 1997, 3, 477–501]. With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric cone C ⁢ Y × ℝ n - k {CY\times\mathbb{R}^{n-k}} ( 2 ≤ k ≤ n {2\leq k\leq n} ) in a non-collapsing metric cone C ⁢ X × ℝ n - k {CX\times\mathbb{R}^{n-k}} obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY. Namely, Y has only one connected component in X.

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension $$n \ge 3$$ . For every bounded open subset $$\Omega \subset M$$ with smooth boundary, we prove that $$\begin{aligned} \int \limits _{\partial \Omega } \left| \frac{\mathrm{H}}{n-1}\right| ^{n-1} \!\!\!\!\!{\mathrm{d}}\sigma \,\,\ge \,\,{\mathrm{AVR}}(g)\,\big |\mathbb {S}^{n-1}\big |, \end{aligned}$$ where $${\mathrm{H}}$$ is the mean curvature of $$\partial \Omega $$ and $${\mathrm{AVR}}(g)$$ is the asymptotic volume ratio of (M, g). Moreover, the equality holds true if and only if $$(M{{\setminus }}\Omega , g)$$ is isometric to a truncated cone over $$\partial \Omega $$ . 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 of Kasue’s non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.

Kähler manifolds of quasi-constant holomorphic sectional curvature and generalized Sasakian space forms

In the present paper, two geometric notions, namely Kahler manifolds of quasi-constant holomorphic sectional curvature and generalized Sasakian space forms, are related to each other, for the first time. Some conditions under which each of these structures induces the other one, are provided here. Several results are obtained on direct products (which are special cases of Naveira’s classification), warped products or hypersurfaces of manifolds and relevant examples are included. A result of Niebergall and Ryan is generalized here. Some necessary and sufficient conditions for Einsteinian hypersurfaces are given at the end.

Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.

Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

Abstract We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a consequence of this sweep-out estimate, there exists an embedded, closed (possibly singular) minimal hypersurface whose volume is bounded in terms of the volume of the manifold.

Splitting of 3-manifolds and rigidity of area-minimising surfaces

In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient manifold are weaker but the assumptions on the surface are considerably more restrictive. We then use our comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature.

Generalized Killing spinors on Einstein manifolds

We study generalized Killing spinors on compact Einstein manifolds with positive scalar curvature. This problem is related to the existence of compact Einstein hypersurfaces in manifolds with parallel spinors, or equivalently, in Riemannian products of flat spaces, Calabi–Yau, hyperkähler, G2 and Spin(7) manifolds.

Free boundary stable hypersurfaces in manifolds with density and rigidity results

Let M be a weighted manifold with boundary ∂M, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in ∂M. As a consequence, we obtain variational characterizations of critical points and second order minima of the weighted area with or without a volume constraint. Moreover, in the compact case, we obtain topological estimates and rigidity properties for free boundary stable and area-minimizing hypersurfaces under certain curvature and boundary assumptions on M. Our results and proofs extend previous ones for Riemannian manifolds (constant densities) and for hypersurfaces with empty boundary in weighted manifolds.

Extrinsic hyperspheres in manifolds with special holonomy

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kähler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.

Solvable Lie algebras are not that hypo

We study a type of left-invariant structure on Lie groups or, equivalently, on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the five-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting $ {\mathfrak{g}^*} = {V_1} \oplus {V_2} $ , and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions for the existence of a hypo structure with a fixed almost-contact form. For nonunimodular Lie algebras, we derive an obstruction to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that admit a hypo structure.

The structure of $${\phi}$$ -stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature

Suppose (N n , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as $${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}$$ , where $${dm=\phi\,dvol(g)}$$ and R(g) is the scalar curvature of (N n , g). In this paper, under a technical assumption on $${\phi}$$ , we prove that $${\phi}$$ -stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane $${\mathbb{C}}$$ or the cylinder $${\mathbb{R}\times\mathbb{S}^1}$$ .

Semi-Riemannian hypersurfaces in manifolds with metric mixed 3-structures

The mixed 3-structures are the counterpart of paraquaternionic structures in odd dimension. A compatible metric with a mixed 3-structure is necessarily semi-Riemann and mixed 3-Sasakian manifolds are Einstein. We investigate the differential geometry of the semi-Riemannian hypersurfaces of co-index both 0 and 1 in a manifold endowed with a mixed 3-structure and a compatible metric.

REDUCED HOLONOMY, HYPERSURFACES AND EXTENSIONS

We study the geometry of hypersurfaces in manifolds with Ricci-flat holonomy group, on which we introduce a G-structure whose intrinsic torsion can be identified with the second fundamental form. The general problem of extending a manifold with such a G-structure so as to invert this construction is open, but results exist in particular cases, which we review. We list the five-dimensional nilmanifolds carrying invariant SU(2)-structures of this type, and present an example of an associated metric with holonomy SU(3).

SU(3)-Structures on Hypersurfaces of Manifolds With G2-Structure

We study SU(3)-structures induced on orientable hypersurfaces of seven-dimensional manifolds with G2-structure. Taking Gray-Hervella types for both structures into account, we relate the type of SU(3)-structure and the type of G2-structure with the shape tensor of the hypersurface. Additionally, we show how to compute the intrinsic SU(3)-torsion and the intrinsic G2-torsion by means of the exterior algebra.

ON HYPERSURFACES OF MANIFOLDS EQUIPPED WITH A HYPERCOSYMPLECTIC 3-STRUCTURE

Hypersurfaces of a Riemannian manifold equipped with a hypercosymplectic 3-structure are studied. Integrability condi- tions for certain distributions on the hypersurface are investigated. Geometry of leaves of certain distribution are also studied. The quaternionic analog of almost complex structures is the almost quaternion (hypercomplex) which is defined by three local (global) al- most complex structures which satisfy the quaternionic relations as the imaginary quaternions satisfy ((3)). Quaternion Kahler manifolds and hyper-Kahler manifolds are special and interesting cases of Riemannian manifolds with almost quaternion and almost hypercomplex structure, respectively. Quaternion Kahler manifolds are Einstein, while hyper- Kahler manifolds are Ricci flat. An almost contact 3-structure was defined by Kuo ((5)) and it is closely related to both almost quaternion and almost hypercomplex structures. Hypersurfaces of manifolds with almost hypercomplex struc- ture inherit naturally three almost contact structures which constitute an almost contact 3-structure. An almost contact metric 3-structure manifold is always (4m + 3)-dimensional. The structural group of the tangent bundle of a (4m + 3)-dimensional manifold equipped with an almost contact 3-structure is reducible to Sp(m) £ I3. In particular, if each almost contact metric structure of an almost contact metric 3-structure is Sasakian, then this structure is called a Sasakian 3-structure. Riemannian manifolds with Sasakian 3-structure

Dirac Operators on Hypersurfaces of Manifolds with Negative Scalar Curvature

We give a sharp extrinsic lower bound for the first eigenvaluesof the intrinsic Dirac operator of certain hypersurfaces boundinga compact domain in a spin manifold of negative scalar curvature.Limiting-cases are characterized by the existence, on the domain,of imaginary Killing spinors. Some geometrical applications, as anAlexandrov type theorem, are given.