Extrinsic Balls Research Articles

Isoperimetric type inequalities on submanifolds in rotation-symmetric spaces

We study the monotonicity formula involving mean curvature for submanifolds in a rotation-symmetric space $N$. Then we get an isoperimetric type inequality on extrinsic balls of submanifolds in $N$ with a sharp constant.

On uniform measures in the Heisenberg group

We study uniform measures in the first Heisenberg group H equipped with the Korányi metric dH. We prove that 1-uniform measures are proportional to the spherical 1-Hausdorff measure restricted to an affine horizontal line, while 2-uniform measures are proportional to spherical 2-Hausdorff measure restricted to an affine vertical line. We also show that each 3-uniform measure which is supported on a vertically ruled surface is proportional to the restriction of spherical 3-Hausdorff measure to an affine vertical plane, and that no quadratic x3-graph can be the support of a 3-uniform measure. According to a result of Merlo, every 3-uniform measure is supported on a quadratic variety; in conjunction with our results, this shows that all 3-uniform measures are proportional to spherical 3-Hausdorff measure restricted to an affine vertical plane. We establish our conclusions by deriving asymptotic formulas for the measures of small extrinsic balls in (H,dH) intersected with smooth submanifolds. The coefficients in our power series expansions involve intrinsic notions of curvature associated to smooth curves and surfaces in H.

Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights

Let P be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight eh. The aim of this paper is twofold. First, by assuming certain control on the h-mean curvature of P, we establish comparisons for the h-capacity of extrinsic balls in P, from which we deduce criteria ensuring the h-parabolicity or h-hyperbolicity of P. Second, we employ functions with geometric meaning to describe submanifolds of bounded h-mean curvature which are confined into some regions of the ambient manifold. As a consequence, we derive half-space and Bernstein-type theorems generalizing previous ones. Our results apply for some relevant h-minimal submanifolds appearing in the singularity theory of the mean curvature flow.

Green functions and the Dirichlet spectrum

This article has results of four types. We show that the first eigenvalue $\lambda\_{1}(\Omega)$ of the weighted Laplacian of a bounded domain with smooth boundary can be obtained by S. Sato's iteration scheme of the Green operator, taking the limit $\lambda\_{1}(\Omega)=\lim\_{k\to \infty} \Vert G^{k}(f)\Vert\_{L^2}/\Vert G^{k+1}(f)\Vert\_{L^2}$ for any $f\in L^{2}(\Omega, \mu)$, $f > 0$. Then, we study the $L^{1}(\Omega, \mu)$-moment spectrum of $\Omega$ in terms of iterates of the Green operator $G$, extending the work of McDonald–Meyers to the weighted setting. As corollary, we obtain the first eigenvalue of a weighted bounded domain in terms of the $L^{1}(\Omega, \mu)$-moment spectrum, generalizing the work of Hurtado–Markvorsen–Palmer. Finally, we study the radial spectrum $\sigma^{\rm rad}(B\_{h}(o,r))$ of rotationally invariant geodesic balls $B\_{h}(o,r)$ of model manifolds. We prove an identity relating the radial eigenvalues of $\sigma^{\rm rad}(B\_{h}(o,r))$ to an isoperimetric quotient, i.e., $\sum 1/\lambda\_{i}^{\rm rad} = \int V(s)/S(s) ds$, $V(s)={\rm vol}(B\_{h}(o,s))$ and $S(s)={\rm vol}(\partial B\_{h}(o,s))$. We then consider a proper minimal surface $M\subset \mathbb{R}^{3}$ and the extrinsic ball $\Omega=M\cap B\_{\mathbb{R}^{3}}(o,r)$. We obtain upper and lower estimates for the series $\sum \lambda\_i^{-2}(\Omega)$ in terms of the volume ${\rm vol}(\Omega)$ and the radius $r$ of the extrinsic ball $\Omega$.

Asymptotically Extrinsic Tamed Submanifolds

We study, from the extrinsic point of view, the structure at infinity of open submanifolds, \(\varphi :M^m \hookrightarrow \mathbb {M}^{n}(\kappa )\) isometrically immersed in the real space forms of constant sectional curvature \(\kappa \le 0\). We shall use the decay of the second fundamental form of the so-called tamed immersions to obtain a description at infinity of the submanifold in the line of the structural results in Greene et al. (Int Math Res Not 1994:364–377, 1994) and Petrunin and Tuschmann (Math Ann 321:775–788, 2001) and an estimation from below of the number of its ends in terms of the volume growth of a special class of extrinsic domains, the extrinsic balls.

Mean curvature, volume and properness of isometric immersions

We explore the relation among volume, curvature and properness of an m m -dimensional isometric immersion in a Riemannian manifold. We show that, when the L p L^p -norm of the mean curvature vector is bounded for some m ≤ p ≤ ∞ m \leq p\leq \infty , and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact 2 2 -Riemannian manifold M M with non-positive Gaussian curvature and finite topology, using the study of the focal points of the transverse Jacobi fields to a geodesic ray in M M . In particular, we have explored the relation between the minimal focal distance of a geodesic ray and the total curvature of an end containing that geodesic ray.

Stable hypersurfaces with zero scalar curvature in Euclidean space

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the radius of the smallest extrinsic ball which contains the domain.

Ends, Fundamental Tones, and Capacities of Minimal Submanifolds Via Extrinsic Comparison Theory

We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotationally symmetric model manifold. Using the asymptotic behavior of the volumes and capacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question.

Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces

We study the topology of (properly) immersed complete minimal surfaces P 2 in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces (see [10]). We present an alternative and unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces (in ℝn and in ℕn(b)), based in the isoperimetric analysis mentioned above. Finally, we show a Chern-Osserman-type equality attained by complete minimal surfaces in the Hyperbolic space with finite total extrinsic curvature.

A spectral Bernstein theorem

We study the spectrum of the Laplace operator of a complete minimal properly immersed hypersurface $M$ in $\R^{n+1}$. (1) Under a volume growth condition on extrinsic balls and a condition on the unit normal at infinity, we prove that $M$ has only essential spectrum consisting of the half line $[0, +\infty)$. This is the case when $\lim_{\tilde{r}\to +\infty}\tilde{r}\kappa_i=0$, where $\tilde{r}$ is the extrinsic distance to a point of $M$ and $\kappa_i$ are the principal curvatures. (2) If the $\kappa_i$ satisfy the decay conditions $|\kappa_i|\leq 1/\tilde{r}$, and strict inequality is achieved at some point $y\in M$, then there are no eigenvalues. We apply these results to minimal graphic and multigraphic hypersurfaces.

Extrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motions in the extrinsic balls, i.e. lower bounds for the time it takes (on average) for Brownian particles to diffuse within the extrinsic ball from a given starting point before they hit the boundary of the extrinsic ball. In those cases, where we may extend our analysis to hold all the way to infinity, we apply a capacity comparison technique to obtain a sufficient condition for the submanifolds to be parabolic, i.e. a condition which will guarantee that any Brownian particle, which is free to move around in the whole submanifold, is bound to eventually revisit any given neighborhood of its starting point with probability 1. The results of this paper are in a rough sense dual to similar results obtained previously by the present authors in complementary settings where we assume that the curvatures are bounded from above.

Stable embedded minimal surfaces bounded by a straight line

We prove that if \({M\subset \mathbb{R}^3}\) is a properly embedded oriented stable minimal surface whose boundary is a straight line and the area of M in extrinsic balls grows quadratically in the radius, then M is a half-plane or half of the classical Enneper minimal surface. This solves a conjecture posed by B. White in Geometric Analysis and the Calculus of Variations for Stefan Hildebrandt, International Press, Somerville, 1996.

On the isoperimetric rigidity of extrinsic minimal balls

We consider an m-dimensional minimal submanifold P and a metric R-sphere in the Euclidean space R n . If the sphere has its center p on P, then it will cut out a well defined connected component of P which contains this center point. We call this connected component an extrinsic minimal R-ball of P. The quotient of the volume of the extrinsic ball and the volume of its boundary is not larger than the corresponding quotient obtained in the space form standard situation, where the minimal submanifold is the totally geodesic linear subspace R m . Here we show that if the minimal submanifold has dimension larger than 3, if P is not too curved along the boundary of an extrinsic minimal R-ball, and if the inequality alluded to above is an equality for the extrinsic minimal ball, then the minimal submanifold is totally geodesic.

Generalized isoperimetric inequalities for extrinsic balls in minimal submanifolds

Article Generalized isoperimetric inequalities for extrinsic balls in minimal submanifolds was published on September 30, 2002 in the journal Journal für die reine und angewandte Mathematik (volume 2002, issue 551).

Curvature Estimates for Stable Minimal Hypersurfaces in R4 and R5

We obtain curvature estimates for certain stable minimalhypersurfaces in R4 and R5without using volume bounds. It follows that if M is acomplete stable minimal hypersurface in R4 orR5, then M is a hyperplane whenM intersects each extrinsic ball in, at most,N-components.

Isoperimetric Inequalities for Extrinsic Balls in Minimal Submanifolds and their Applications

S.-Y. Cheng, P. Li and S.-T. Yau proved comparison theorems for the volume of extrinsic balls in minimal submanifolds of space forms. These results were extended by S. Markvorsen for minimal submanifolds of a riemannian manifold with just an upper bound on the sectional curvature. In the paper an isoperimetric inequality for extrinsic balls in minimal submanifolds of a riemannian manifold N with sectional curvatures bounded from above by a non-positive constant is found. As a corollary of this result an alternative proof is obtained of the comparison for the volume of extrinsic balls stated by the preceding authors, but now the equality case is characterized when the upper bound for the sectional curvatures of the ambient manifold is strictly negative. Finally, when the sectional curvatures of N are bounded from above for any constant (positive or negative), it is proved that the ∞-isoperimetric quotient of the extrinsic balls is bounded from below by the mean curvature of the geodesic spheres in the m-dimensional real space forms.

On the volume of a small extrinsic ball in a hypersurface of the hyperbolic space

On the volume of a small extrinsic ball in a hypersurface of the hyperbolic space

Volume of a small Extrinsic Ball in a Submanifold

Volume of a small Extrinsic Ball in a Submanifold