Mean Convex Boundary Research Articles

The first Dirichlet eigenvalue and the width

For a geodesic ball with non-negative Ricci curvature and mean convex boundary, it is known that the first Dirichlet eigenvalue of this geodesic ball has a sharp lower bound in term of its radius. We show a quantitative explicit inequality, which bounds the width of geodesic ball in terms of the spectral gap between the first Dirichlet eigenvalue and the corresponding sharp lower bound.

Uryson width of three dimensional mean convex domain with non-negative Ricci curvature

We prove that for every three dimensional manifold with nonnegative Ricci curvature and strictly mean convex boundary (non-empty), there exists a Morse function so that each connected component of its level sets has a uniform diameter bound, which depends only on the lower bound of mean curvature. This gives an upper bound of Uryson 1-width for those three manifolds with boundary.

The location of hot spots and other extremal points

In a domain of the Euclidean space, we estimate from below the distance to the boundary of global maximum points of solutions of elliptic and parabolic equations with homogeneous Dirichlet boundary values. As reference cases, we consider the torsional rigidity function of a bar, the first mode of a vibrating membrane, and the temperature of a heat conductor grounded to zero at the boundary. Our main results are presented for domains with mean convex boundary and compare that distance to the inradius of the relevant domain. For the torsional rigidity function, the obtained bound only depends on the space dimension. The more general case of a boundary which is not mean convex is also considered. However, in this case the estimates also depend on some geometrical quantities such as the diameter and the radius of the largest exterior osculating ball to the relevant domain, or the minimum of the mean curvature of the boundary. For the first mode, the relevant bound only depends on the space dimension, as well. Moreover, it largely improves on an earlier estimate obtained for convex domains by the first author and co-authors. The bound related to the temperature depends on time and the initial distribution of temperature. Such a bound is substantially consistent with what one obtains in the stationary situation. The methods employed are based on elementary arguments and existing literature, and can be extended to other situations that entail quasilinear equations, isotropic and anisotropic, and also certain classes of semilinear equations.

A two-piece property for free boundary minimal surfaces in the ball

We prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean 3 3 -ball in exactly two connected surfaces. We also show that if a region in the ball has mean convex boundary and contains a nullhomologous diameter, then this region is a closed halfball. Moreover, we prove the regularity at the corners of currents minimizing a partially free boundary problem by following ideas by Grüter and Simon. Our first result gives evidence to a conjecture by Fraser and Li.

Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

The Mass of an Asymptotically Hyperbolic Manifold with a Non-compact Boundary

We define a mass-type invariant for asymptotically hyperbolic manifolds with a non-compact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application, we show that any such manifold which is Einstein and either has a totally geodesic boundary or is conformally compact and has a mean convex boundary is isometric to the hyperbolic half-space.

Determining a Riemannian metric from minimal areas

We prove that if (M,g) is a topological 3-ball with a C4-smooth Riemannian metric g, and mean-convex boundary ∂M, then knowledge of least areas circumscribed by simple closed curves γ⊂∂M uniquely determines the metric g, under some additional geometric assumptions. These are that g is either a) C3-close to Euclidean or b) satisfies much weaker geometric conditions which hold when the manifold is to a sufficient degree either thin, or straight.In fact, the least area data that we require is for a much smaller subset of curves γ⊂∂M. We also prove a local result: Given a point p∈∂M where ∂M is mean convex, we provide a reconstruction of the metric near p, given suitable area data near p only.The proofs rely on finding the metric along a continuous sweep-out of M by area-minimizing surfaces; they bring together ideas from the 2D-Calderón inverse problem, minimal surface theory, and the careful analysis of a system of pseudo-differential equations.

Bubbling analysis and geometric convergence results for free boundary minimal surfaces

We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient dimension less than eight and applicable to possibly improper limit hypersurfaces. In dimension three, this identity can be combined with the Gauss-Bonnet theorem to provide a constraint relating the topology of the free boundary minimal surfaces in a converging sequence, of their limit, and of the bubbles or half-bubbles that occur as blow-up models. We present various geometric applications of these tools, including a description of the behaviour of index one free boundary minimal surfaces inside a 3-manifold of non-negative scalar curvature and strictly mean convex boundary. In particular, in the case of compact, simply connected, strictly mean convex domains in ℝ 3 unconditional convergence occurs for all topological types except the disk and the annulus, and in those cases the possible degenerations are classified.

Free Boundary Hypersurfaces with Non-positive Yamabe Invariant in Mean Convex Manifolds

We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface $$\Sigma $$ with non-positive Yamabe invariant in a Riemannian n-manifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that $$\Sigma $$ is locally volume-minimizing in a manifold $$M^n$$ with scalar curvature bounded from below by a non-positive constant and mean convex boundary, we conclude that locally M splits along $$\Sigma $$ . In the case that the scalar curvature of M is at least $$-n(n-1)$$ and $$\Sigma $$ locally minimizes a certain functional inspired by works of Yau [35] and Andersson-Galloway [4], a neighbourhood of $$\Sigma $$ in M is isometric to $$((-\varepsilon , \varepsilon ) \times \Sigma ,\mathrm{{d}}t^{2}+e^{2t}g)$$ , where g is Ricci flat with totally geodesic boundary.

Eigenvalue estimates of Reilly type in product manifolds and eigenvalue comparison for strip domains

In the first part we derive sharp upper bounds of Reilly type for three kinds of eigenvalues in product manifolds Rk×Mn+1−k for any complete Riemannian manifold M. The eigenvalues include the first Laplacian eigenvalue on mean convex closed hypersurfaces, the first Steklov eigenvalue on domains with mean convex boundary, and the first Hodge Laplacian eigenvalue on closed hypersurfaces with certain convexity condition. In the second part, we prove a comparison result between the first Steklov eigenvalue of a strip domain in space forms and that of the corresponding warped product manifold.

Comparison theorems for manifolds with mean convex boundary

Let Mn be an n-dimensional Riemannian manifold with boundary ∂M. Assuming that Ricci curvature is bounded from below by (n - 1)k, for k ∈ ℝ, we give a sharp estimate of the upper bound of ρ(x) = d (x, ∂M), in terms of the mean curvature bound of the boundary. When ∂M is compact, the upper bound is achieved if and only if M is isometric to a disk in space form. A Kähler version of estimation is also proved. Moreover, we prove a Laplacian comparison theorem for distance function to the boundary of Kähler manifold and also estimate the first eigenvalue of the real Laplacian.

Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru (Splitting of 3-manifolds and rigidity of area-minimizing surfaces, arXiv:1107.5346, 2011). We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.

F-Minimal Surface and Manifold with Positive m-Bakry–Émery Ricci Curvature

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the $f$-Laplacian on compact manifold with positive $m$-Bakry-\'{E}mery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the $n$-shpere, or the $n$-dimensional hemisphere. Finally, for compact manifold with positive $m$-Bakry-\'{E}mery Ricci curvature and $f$-mean convex boundary, we prove an upper bound for the distance function to the boundary, and the upper bound is achieved if only if the manifold is isometric to an Euclidean ball.

A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\partial M$. Assume that the mean curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for some positive constant $k$. In this paper, we prove that the distance function $d$ to the boundary $\partial M$ is bounded from above by $\frac{1}{k}$ and the upper bound is achieved if and only if $M$ is isometric to an $n$-dimensional Euclidean ball of radius $\frac{1}{k}$.

Extension of a theorem of Shi and Tam

In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79–125, 2002): Let (Ω, g) be an n-dimensional (n ≥ 3) compact Riemannian manifold, spin when n > 7, with non-negative scalar curvature and mean convex boundary. If every boundary component Σi has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface \({{\hat \Sigma}_i \subset \mathbb{R}^n}\), then $$ \int\limits_{\Sigma_i} H d \sigma \le \int\limits_{{\hat \Sigma}_i} \hat{H} d {\hat \sigma} $$ where H is the mean curvature of Σi in (Ω, g), \({\hat{H}}\) is the Euclidean mean curvature of \({{\hat \Sigma}_i}\) in \({\mathbb{R}^n}\), and where d σ and \({d {\hat \sigma}}\) denote the respective volume forms. Moreover, equality holds for some boundary component Σi if, and only if, (Ω, g) is isometric to a domain in \({\mathbb{R}^n}\). In the proof, we make use of a foliation of the exterior of the \({\hat \Sigma_i}\)’s in \({\mathbb{R}^n}\) by the \({\frac{H}{R}}\)-flow studied by Gerhardt (J Differ Geom 32:299–314, 1990) and Urbas (Math Z 205(3):355–372, 1990). We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in Shi and Tam (J Differ Geom 62:79–125, 2002).

A Volumetric Penrose Inequality for Conformally Flat Manifolds

We consider asymptotically flat Riemannian manifolds with nonnegative scalar curvature that are conformal to $\R^{n}\setminus \Omega, n\ge 3$, and so that their boundary is a minimal hypersurface. (Here, $\Omega\subset \R^{n}$ is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by $(V/\beta_{n})^{(n-2)/n}$, where $V$ is the Euclidean volume of $\Omega$ and $\beta_{n}$ is the volume of the Euclidean unit $n$-ball. This gives a partial proof to a conjecture of Bray and Iga \cite{brayiga}. Surprisingly, we do not require the boundary to be outermost.

Extrinsic Killing spinors

Under intrinsic and extrinsic curvature assumptions on a Riemannian spin manifold and its boundary, we show that there is an isomorphism between the restriction to the boundary of parallel spinors and extrinsic Killing spinors of non-negative Killing constant. As a corollary, we prove that a complete Ricci-flat spin manifold with mean-convex boundary isometric to a round sphere, is necessarily a flat disc.

Eigenvalue Boundary Problems for the Dirac Operator

On a compact Riemannian spin manifold with mean-convex boundary, we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental Dirac operator including the (global) APS condition and a Riemannian version of the (local) MIT bag condition. We show that Friedrich's inequality for the eigenvalues of the Dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems. More precisely, we prove that, for both the APS and the MIT conditions, the equality cannot be achieved, and for the other two conditions, the equality characterizes respectively half-spheres and domains bounded by minimal hypersurfaces in manifolds carrying non-trivial real Killing spinors.

Minimal surfaces, spatial topology and singularities in space-time

Using an approach first advocated by Gannon (1975, 1976) and recent results of Meeks, Simon and Yau (1982) on the existence of compact minimal surfaces some new results are obtained relating non-trivial spatial topology to the occurrence of singularities in space-time. For example, it is shown that if V3 is a contracting body with mean-convex boundary homeomorphic to a two-sphere in a space-time M4 obeying appropriate curvature and causality assumptions, then either V3 is a three-cell or M4 is non-space-like geodesically incomplete.