Bounded Mean Curvature Research Articles

Isoperimetric Residues and a Mesoscale Flatness Criterion for Hypersurfaces with Bounded Mean Curvature

Isoperimetric Residues and a Mesoscale Flatness Criterion for Hypersurfaces with Bounded Mean Curvature

Mean Exit Times from Submanifolds with Bounded Mean Curvature

Mean Exit Times from Submanifolds with Bounded Mean Curvature

Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimension three

abstract: We prove a priori interior curvature estimates for hypersurfaces of prescribing scalar curvature equations in $\mathbb{R}^{3}$. The method is motivated by the integral method of Warren and Yuan. The new observation here is that we construct a ``Lagrangian'' graph which is a submanifold of bounded mean curvature if the graph function of a hypersurface satisfies a scalar curvature equation.

Correction: A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary

Correction: A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary

On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

ON THE GEOMETRY OF SPACELIKE MEAN CURVATURE FLOW SOLITONS IMMERSED IN A GRW SPACETIME

Abstract We investigate geometric aspects of complete spacelike mean curvature flow solitons of codimension one in a generalized Robertson–Walker (GRW) spacetime $-I\times _{f}M^n$ , with base $I\subset \mathbb R$ , Riemannian fiber $M^n$ and warping function $f\in C^\infty (I)$ . For this, we apply suitable maximum principles to guarantee that such a mean curvature flow soliton is a slice of the ambient space and to obtain nonexistence results concerning these solitons. In particular, we deal with entire graphs constructed over the Riemannian fiber $M^n$ , which are spacelike mean curvature flow solitons, and we also explore the geometry of a conformal vector field to establish topological and further rigidity results for compact (without boundary) mean curvature flow solitons in a GRW spacetime. Moreover, we study the stability of spacelike mean curvature flow solitons with respect to an appropriate stability operator. Standard examples of spacelike mean curvature flow solitons in GRW spacetimes are exhibited, and applications related to these examples are given.

A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary

A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary

On foliations of bounded mean curvature on closed three-dimensional Riemannian manifolds

The notion of systole of a foliation sys(ℱ) on an arbitrary foliated closed Riemannian manifold (M,ℱ) is introduced. A lower bound on sys(ℱ) of a bounded mean curvature foliation is given. As a corollary we prove that the number of Reeb components of a bounded mean curvature foliation on a closed oriented Riemannian 3-manifold M is bounded above by a constant depending on the volume, the radius of injectivity, and the maximum value of the sectional curvature of the manifold M.

Classification of convex ancient free boundary mean curvature flows in the ball

Classification of convex ancient free boundary mean curvature flows in the ball

Existence of mean curvature flow singularities with bounded mean curvature

In “Curvature blow-up in perturbations of minimal cones evolving by mean curvature flow,” Velázquez constructed a countable collection of mean curvature flow solutions in RN in every dimension N≥8. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. In contrast, we confirm here that, in every dimension N≥8, infinitely many of these solutions have uniformly bounded mean curvature.

A priori estimates versus arbitrarily large solutions for fractional semi-linear elliptic equations with critical Sobolev exponent

We study positive solutions to the fractional semi-linear elliptic equation $${\left( { - \Delta } \right)^\sigma }u = K\left( x \right){u^{{{n + 2\sigma } \over {n - 2\sigma }}\,}}\,\,\,{\rm{in}}\,{B_2}\backslash \left\{ 0 \right\}$$ with an isolated singularity at the origin, where K is a positive function on B2, the punctured ball B2 {0} ⊂ ℝn with n ⩾ 2, σ ∈ (0, 1), and (−Δ)σ is the fractional Laplacian. In lower dimensions, we show that for any K ∈ C1(B2), a positive solution u always satisfies that u(x) ⩽ C∣x∣−(n−2σ)/2 near the origin. In contrast, we construct positive functions K ∈ C1(B2) in higher dimensions such that a positive solution u could be arbitrarily large near the origin. In particular, these results also apply to the prescribed boundary mean curvature equations on $${\mathbb{B}^{n + 1}}$$ .

A barrier principle at infinity for varifolds with bounded mean curvature

Our work investigates varifolds $\Sigma \subset M$ in a Riemannian manifold, with arbitrary codimension and bounded mean curvature, contained in an open domain $\Omega$. Under mild assumptions on the curvatures of $M$ and on $\partial \Omega$, also allowing for certain singularities of $\partial \Omega$, we prove a barrier principle at infinity, namely we show that the distance of $\Sigma$ to $\partial \Omega$ is attained on $\partial \Sigma$. Our theorem is a consequence of sharp maximum principles at infinity on varifolds, of independent interest.

On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature

In this paper, we study the principal eigenvalue $\mu(\mathscr{F}_k^-,E)$ of the fully nonlinear operator \[ \mathscr{F}_k^-[u] = \mathcal{P}_k^-(\nabla^2 u) - h |\nabla u| \] on a set $E \Subset \mathbb{R}^n$, where $h \in [0,\infty)$ and $\mathcal{P}_k^-(\nabla^2 u)$ is the sum of the smallest $k$ eigenvalues of the Hessian $\nabla^2 u$. We prove a lower estimate for $\mu(\mathscr{F}_k^-,E)$ in terms of a generalized Hausdorff measure $\mathscr{H}_\Psi(E)$, for suitable $\Psi$ depending on $k$, moving some steps in the direction of the conjecturally sharp estimate \[ \mu(\mathscr{F}_k^-,E) \ge C \mathscr{H}^k(E)^{-2/k}. \] The theorem is used to study the spectrum of bounded submanifolds in $\mathbb{R}^n$, improving on our previous work in the direction of a question posed by S.T. Yau. In particular, the result applies to solutions of Plateau's problem for CMC surfaces.

The Nirenberg problem on high dimensional half spheres: the effect of pinching conditions

In this paper we study the Nirenberg problem on standard half spheres (mathbb {S}^n_+,g), , n ge 5, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: (P)-Δgu+n(n-2)4u=Kun+2n-2,u>0inS+n,∂u∂ν=0on∂S+n.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} (\\mathcal {P}) \\quad {\\left\\{ \\begin{array}{ll} -\\Delta _{g} u \\, + \\, \\frac{n(n-2)}{4} u \\, = K \\, u^{\\frac{n+2}{n-2}},\\, u > 0 &{}\\quad \\text{ in } \\mathbb {S}^n_+, \\\\ \\frac{\\partial u}{\\partial \\nu }\\, =\\, 0 &{}\\quad \\text{ on } \\partial \\mathbb {S}^n_+. \\end{array}\\right. } \\end{aligned}$$\\end{document}where K in C^3(mathbb {S}^n_+) is a positive function. This problem has a variational structure but the related Euler–Lagrange functional J_K lacks compactness. Indeed it admits critical points at infinity, which are limits of non compact orbits of the (negative) gradient flow. Through the construction of an appropriate pseudogradient in the neighborhood at infinity, we characterize these critical points at infinity, associate to them an index, perform a Morse type reduction of the functional J_K in their neighborhood and compute their contribution to the difference of topology between the level sets of J_K, hence extending the full Morse theoretical approach to this non compact variational problem. Such an approach is used to prove, under various pinching conditions, some existence results for (mathcal {P}) on half spheres of dimension n ge 5.

Second order rectifiability of varifolds of bounded mean curvature

We prove that the support of an m dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered {mathscr {H}}^{m} almost everywhere by a countable union of m dimensional submanifolds of class {mathcal {C}}^{2} . The {mathcal {C}}^{2} -regularity of the submanifolds is optimal.

Geometric estimates on weighted p-fundamental tone and applications to the first eigenvalue of submanifolds with bounded mean curvature

ABSTRACT This paper generalizes to the context of smooth metric measure spaces and submanifolds with negative sectional curvatures some well-known geometric estimates on the p-fundamental tone by using vector fields satisfying a positive divergence condition. Choosing the vector field to be the gradient of an appropriately chosen distance function yields generalised McKean estimates whilst other choices of vector fields yield new geometric estimates generalising certain results of Lima et al. (Nonlinear Anal. 2010;72:771–781). We also obtain a lower bound on the spectrum of the weighted p-Laplacian on a complete noncompact smooth metric space with the underlying space being a submanifold with bounded mean curvature in the hyperbolic space form of constant negative sectional curvature generalising results of Du and Mao (J Math Anal Appl. 2017;456:787–795).

First Robin eigenvalue of the p-Laplacian on Riemannian manifolds

We consider the first Robin eigenvalue $$\lambda _p(M,\alpha )$$ for the p-Laplacian on a compact Riemannian manifold M with nonempty smooth boundary, with $$\alpha \in \mathbb {R}$$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $$\lambda _p(M,\alpha )$$ . Secondly, when $$\alpha >0$$ we establish sharp lower bound of $$\lambda _p(M,\alpha )$$ in terms of dimension, inradius, Ricci curvature lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. The lower bound becomes an upper bound when $$\alpha <0$$ . Our results cover corresponding comparison theorems for the first Dirichlet eigenvalue of the p-Laplacian when letting $$\alpha \rightarrow +\infty $$ .

The singular structure and regularity of stationary varifolds

If one considers an integral varifold I^m\subseteq M with bounded mean curvature, and if S^k(I)\equiv\{x\in M: no tangent cone at x is k+1 -symmetric} is the standard stratification of the singular set, then it is well known that \mathrm {dim} S^k\leq k . In complete generality nothing else is known about the singular sets S^k(I) . In this paper we prove for a general integral varifold with bounded mean curvature, in particular a stationary varifold, that every stratum S^k(I) is k -rectifiable. In fact, we prove for k -a.e. point x\in S^k that there exists a unique k -plane V^k such that every tangent cone at x is of the form V\times C for some cone C . In the case of minimizing hypersurfaces I^{n-1}\subseteq M^n we can go further. Indeed, we can show that the singular set S(I) , which is known to satisfy \mathrm {dim} S(I)\leq n-8 , is in fact n-8 rectifiable with uniformly finite n-8 measure. An effective version of this allows us to prove that the second fundamental form A has a priori estimates in L^7_{\mathrm {weak}} on I , an estimate which is sharp as |A| is not in L^7 for the Simons cone. In fact, we prove the much stronger estimate that the regularity scale r_I has L^7_{weak} -estimates. The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications S^k_{\epsilon,r} and S^k_{\epsilon}\equiv S^k_{\epsilon,0} . Roughly, x\in S^k_{\epsilon}\subseteq I if no ball B_r(x) is \epsilon -close to being k+1 -symmetric. We show that S^k_\epsilon is k -rectifiable and satisfies the Minkowski estimate \mathrm {Vol}(B_r\,S_\epsilon^k)\leq C_\epsilon r^{n-k} . The proof requires a new L^2 -subspace approximation theorem for integral varifolds with bounded mean curvature, and a W^{1,p} -Reifenberg type theorem proved by the authors in [NVa].

Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation

We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of “lower energy” solutions to the above equation fails when the dimension of the manifold is not less than 62.

Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature

In this paper we deal with a class of varieties of bounded mean curvature in the viscosity sense that has the remarkable property to contain the blow up sets of all sequences of varifolds whose mean curvatures are uniformly bounded and whose boundaries are uniformly bounded on compact sets. We investigate the second-order properties of these varieties, obtaining results that are new also in the varifold’s setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, a property that allows to prove a Coarea-type formula for their generalized Gauss map. Then we use this formula to extend a sharp geometric inequality of Almgren and the associated soap bubble theorem. As a consequence of the geometric inequality we obtain sufficient conditions to conclude that the area-blow-up set is empty for sequences of varifolds whose first variation is controlled.