Elliptic Regularization of the Isometric Immersion Problem

In an earlier article we obtain a sharp inequality for an arbitrary isometric immersion from a Riemannian manifold admitting a Riemannian submersion with totally geodesic fibres into a unit sphere. In this article we investigate the immersions which satisfy the equality case of the inequality. As a by-product, we discover a new characterisation of Cartan hypersurface in S4.

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Unique isometric immersion into hyperbolic space

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171293000900 ◽

1993 ◽

Vol 16 (4) ◽

pp. 725-732 ◽

Cited By ~ 1

Author(s):

M. Beltagy

Keyword(s):

Hyperbolic Space ◽

Isometric Immersion ◽

Isometric Immersions ◽

Gauss Maps

A result concerning equivalence of isometric immersions with related Gauss maps into hyperbolic space has been established.

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On Isometric Immersion of Three-Dimensional Geometries $$\widetilde{SL}_2$$ , Nil, and Sol into a Four-Dimensional Space of Constant Curvature

Ukrainian Mathematical Journal ◽

10.1007/s11253-005-0206-7 ◽

2005 ◽

Vol 57 (3) ◽

pp. 509-516 ◽

Cited By ~ 3

Author(s):

L. A. Masal'tsev

Keyword(s):

Constant Curvature ◽

Isometric Immersion ◽

Dimensional Space ◽

Three Dimensional ◽

Space Of Constant Curvature

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Normal and osculating maps for submanifolds of RN

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500024252 ◽

1990 ◽

Vol 114 (1-2) ◽

pp. 39-55 ◽

Cited By ~ 3

Author(s):

I. Cattaneo Gasparini ◽

G. Romani

Keyword(s):

Isometric Immersion ◽

Normal Bundle ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Gauss Map ◽

Necessary And Sufficient Conditions ◽

Totally Geodesic ◽

Precise Formulation ◽

Parallel Case ◽

Necessary And Sufficient

SynopsisLet Mn be a manifold supposed “nicely curved” isometrically immersed in ℝn+p. Starting from a generalised Gauss map associated to the splitting of the normal bundle defined from the values of the fundamental forms of M of order k (k ≧ 0), we give necessary and sufficient conditions for the map to be totally geodesic and harmonic . For k = 0 is the classical Gauss map and our formula reduces to Ruh–Vilm's formula with a more precise formulation due to the consideration of the splitting of the normal bundle.We also give necessary conditions for M, supposed complete, to admit an isometric immersion with . This theorem generalises a theorem of Vilms on the manifolds with second fundamental forms parallel (case k = 0). The result is interesting as the class of manifolds satisfying the condition is larger than the class of manifolds satisfying .

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The free-boundary Brakke flow

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2017-0053 ◽

2020 ◽

Vol 2020 (758) ◽

pp. 95-137 ◽

Cited By ~ 1

Author(s):

Nick Edelen

Keyword(s):

Free Boundary ◽

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow ◽

Monotonicity Formula ◽

Local Regularity ◽

Regularity Theorem ◽

Elliptic Regularization ◽

Barrier Surface ◽

Boundary Mean Curvature

AbstractWe develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must “pop” upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula valid at all points, and use this to adapt the local regularity theorem of White [23] to the free-boundary setting. Using Ilmanen’s elliptic regularization procedure [10], we prove existence of free-boundary Brakke flows.

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