Homotopy of area decreasing maps by mean curvature flow

Savas-Halilaj, Andreas; Smoczyk, Knut

doi:10.1016/j.aim.2014.01.014

Cited by 17 publications

(22 citation statements)

References 18 publications

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“…Clutterbuck and Schnürer [5] considered graphical solutions to mean curvature flow and obtained a stability result for homothetically expanding solutions coming out of cones of positive mean curvature. It is expected that similar results hold for the mean curvature flow in higher codimension of entire graphs generated by contractions and area decreasing maps as studied in [26][27][28].…”

Section: Introductionsupporting

confidence: 56%

Self-Expanders of the Mean Curvature Flow

Smoczyk

2021

Vietnam J. Math.

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We study self-expanding solutions $M^{m}\subset \mathbb {R}^{n}$ M m ⊂ ℝ n of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function |A|2/|H|2 attains a local maximum, where A denotes the second fundamental form and H the mean curvature vector of M. If the principal normal ξ = H/|H| is parallel in the normal bundle, then a similar result holds in higher codimension for the function |Aξ|2/|H|2, where Aξ is the second fundamental form with respect to ξ. As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension 2 any mean convex self-expander that is asymptotic to a cone must be strictly convex.

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Section: Introductionsupporting

confidence: 56%

Self-Expanders of the Mean Curvature Flow

Smoczyk

2021

Vietnam J. Math.

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“…If z = 0 on W 1 , then e 1 (z) = e 3 (z) = 0 on W 1 . Using (43), we easily conclude that the function χ is constant on W 1 . Consequently, (45) gives τ = 0 on W 1 which is a contradiction.…”

Section: Proofmentioning

confidence: 89%

“…In this section, we briefly discuss the geometry of maps between Riemannian manifolds following the notation in [41][42][43][44].…”

Section: Maps Between Riemannian Manifoldsmentioning

confidence: 99%

Rigidity of the Hopf fibration

Markellos

Savas-Halilaj

2021

Calc. Var.

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In this paper, we study minimal maps between euclidean spheres. The Hopf fibrations provide explicit examples of such minimal maps. Moreover, their corresponding graphs have second fundamental form of constant norm. We prove that a minimal submersion from S 3 to S 2 whose Gauss map satisfies a suitable pinching condition must be weakly conformal with totally geodesic fibers. As a consequence, we obtain that an equivariant minimal submersion from S 3 to S 2 coincides with the Hopf fibration. Furthermore, we prove that a minimal map f : U ⊂ S 3 → S 2 with constant singular values and constant norm of the second fundamental form is either constant or, up to isometries, coincides with the Hopf fibration.

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“…For example, if f : M → N is strictly area-decreasing, M and N are space forms with dim M ≥ 2, and their sectional curvatures satisfy sec M ≥ | sec N | , sec M + sec N > 0 , Wang and Tsui proved long-time existence of the graphical mean curvature flow and convergence of f to a constant map [19]. Subsequently, the curvature assumptions on the manifolds were relaxed by Lee and Lee [9] and recently by Savas-Halilaj and Smoczyk [15].…”

Section: Introductionmentioning

confidence: 99%

Evolution of Area-Decreasing Maps Between Two-Dimensional Euclidean Spaces

Lubbe¹

2018

J Geom Anal

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We consider the mean curvature flow of the graph of a smooth map f : R 2 → R 2 between two-dimensional Euclidean spaces. If f satisfies an areadecreasing property, the solution exists for all times and the evolving submanifold stays the graph of an area-decreasing map ft. Further, we prove uniform decay estimates for the mean curvature vector of the graph and all higher-order derivatives of the corresponding map ft.

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Homotopy of area decreasing maps by mean curvature flow

Cited by 17 publications

References 18 publications

Self-Expanders of the Mean Curvature Flow

Self-Expanders of the Mean Curvature Flow

Rigidity of the Hopf fibration

Evolution of Area-Decreasing Maps Between Two-Dimensional Euclidean Spaces

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