Isoperimetric Inequalities for Extrinsic Balls in Minimal Submanifolds and their Applications

Palmer, Vicente

doi:10.1112/s0024610799007760

Cited by 33 publications

(48 citation statements)

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“…We note that for most minimally immersed submanifolds P m in the flat Euclidean spaces R n with the obvious choice of comparison model space, M m W = R m , W (r ) = r , we have (see [11,20]):…”

Section: Resultsmentioning

confidence: 99%

Torsional rigidity of submanifolds with controlled geometry

2008

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We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P m with controlled radial mean curvature in ambient Riemannian manifolds N n with a pole p and with sectional curvatures bounded from above and from below, respectively. These bounds are given in terms of the torsional rigidities of corresponding Schwarz-symmetrization of the domains in warped product model spaces. Our main results are obtained using methods from previously established isoperimetric inequalities, as found in, e.g., Markvorsen and Palmer (Proc Lond Math Soc 93:253-272, 2006; Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below, p. 39, preprint, 2007). As in that paper we also characterize the geometry of those situations in which the bounds for the torsional rigidity are actually attained and study the behavior at infinity of the so-called geometric average of the mean exit time for Brownian motion. Mathematics Subject Classification (2000)Primary 53C42 · 58J65 · 35J25 · 60J65 S.

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Section: Resultsmentioning

confidence: 99%

Torsional rigidity of submanifolds with controlled geometry

2008

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“…In passing we note that when equality is attained in (1.7) for some fixed radius R, and when the ambient space N n is the hyperbolic space H n (b), b < 0, then the minimal submanifold itself is a totally geodesic hyperbolic subspace H m (b) of H n (b), see [Pa2]. Thus, in analogy with the St. Venant torsion problem -and in analogy with the classical isoperimetric problem itself -we also obtain strong rigidity conclusions from equalities in these isoperimetric estimates.…”

Section: Introductionmentioning

confidence: 99%

“…[Pa2,MP1,MP4,HMP,MP5]. In these works we use R-balls and R-spheres in tailor made rotationally symmetric (warped product) model spaces M m w as comparison objects.…”

Section: Introductionmentioning

confidence: 99%

“…The simplest settings considered are given by the minimal submanifolds P m in real space forms K n (b) of constant sectional curvature b 0. In these specific cases we have the following isoperimetric inequalities, see [CLY,Ma1,Ma2,Pa2,MP1]: R denote, respectively, the geodesic R-ball and the geodesic R-sphere in the real space form K n (b).…”

Section: Introductionmentioning

confidence: 99%

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Comparison of Exit Moment Spectra for Extrinsic Metric Balls

2011

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ABSTRACT. We prove explicit upper and lower bounds for the L 1 -moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds P m in ambient Riemannian spaces N n . We assume that P and N both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as viewed from a pole in N. The bounds for the exit moment spectra are given in terms of the corresponding spectra for geodesic metric balls in suitably warped product model spaces. The bounds are sharp in the sense that equalities are obtained in characteristic cases. As a corollary we also obtain new intrinsic comparison results for the exit time spectra for metric balls in the ambient manifolds N n themselves.

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“…In fact, there are in the literature a number of works where other geometric quantities defined on the extrinsic balls and spheres, such like its volume, have been used for similar purposes, (see e.g. the paper [2], where a characterization of totally geodesic embeddings is obtained from an asymptotic formula for the volume of an extrinsic ball of small radius in a submanifold of the n-dimensional euclidean space, or the papers [3] and [4], which deal with the behaviour of the quotient between the volumes of the extrinsic…”

Section: Introductionmentioning

confidence: 99%

Mean curvature of extrinsic spheres in submanifolds of real space forms

Palmer

Piñero

2004

Arch. Math.

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Given a submanifold P m with the Hilbert-Schmidt norm of its second fundamental form bounded from above, in a real space form of constant curvature b, K n (b), we have obtained a lower bound for the norm of the mean curvature normal vector field of extrinsic spheres with sufficiently small radius in P m in terms of the mean curvature of the geodesic spheres in K m (b), with same radius, and the mean curvature of P m . Introduction.In the paper [5] it was obtained a lower bound for the norm of the mean curvature normal vector field of the boundary of some well-characterized domains, the extrinsic balls, defined in a hypersurface of the real space forms with constant sectional curvature b, that we shall denote as K n (b), in terms of the mean curvature of the geodesic spheres of these space forms and the mean curvature of the hypersurface. The purpose of this work is to present the same lower bound for the norm of the mean curvature normal vector field defined on the boundary of the extrinsic balls, when these extrinsic domains are placed on a submanifold with any codimension in the real space forms, and we have the additional restriction that the norm of the second fundamental form of this submanifold is bounded from above, i.e., the submanifold is not too extrinsically curved.This objective takes place in the more general setting which deals with to what extent information about the extrinsic curvatures of the extrinsic spheres, (the boundary of the extrinsic balls), in a submanifold P m can give us an idea about the way the whole submanifold P m is immersed in the ambient manifold or about some other global geometric properties of P m . In fact, there are in the literature a number of works where other geometric quantities defined on the extrinsic balls and spheres, such like its volume, have been used for similar purposes, (see e.g. the paper [2], where a characterization of totally geodesic embeddings is obtained from an asymptotic formula for the volume of an extrinsic ball of small radius in a submanifold of the n-dimensional euclidean space, or the papers [3] and [4], which deal with the behaviour of the quotient between the volumes of the extrinsic

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Isoperimetric Inequalities for Extrinsic Balls in Minimal Submanifolds and their Applications

Cited by 33 publications

References 0 publications

Torsional rigidity of submanifolds with controlled geometry

Torsional rigidity of submanifolds with controlled geometry

Comparison of Exit Moment Spectra for Extrinsic Metric Balls

Mean curvature of extrinsic spheres in submanifolds of real space forms

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