Evolution of contractions by mean curvature flow

Savas-Halilaj, Andreas; Smoczyk, Knut

doi:10.1007/s00208-014-1090-y

Cited by 16 publications

(18 citation statements)

References 12 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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“…for some positive number 0 > σ , is homotopic to a constant map. Other vanishing theorems can be found in [21] and [30]. In the fourth section of our paper we prove that any harmonic contraction mapping between a compact Riemannian manifold ( with respect to the local coordinates of U and U .…”

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confidence: 80%

“…To summarize the above observations (see also [13] and [18]) we note that until today all known results on harmonic maps between Riemannian manifolds are based in an essential way on the assumption that the target manifold ( ) [21]). In particular, the map f is called strictly…”

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confidence: 90%

Vanishing theorems for harmonic mappings into non-negatively curved manifolds and their applications

Степанов

Tsyganok

2017

manuscripta math.

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So far, all known results on harmonic maps between Riemannian manifolds are based in an essential way on the assumption that the target manifold has non-positive sectional curvature. In our paper we develop a theory of harmonic mappings of Riemannian manifolds into non-negatively curved Riemannian manifolds and give the geometric applications of our results to the theory of holomorphic maps of almost Kählerian manifolds and to addressing the "prescribed Ricci curvature problem". Keywords Riemannian manifold · Harmonic map · Vanishing theoremMathematical Subject Classification 53C20 Introduction and resultsIn this section we focus our attention on the applications of the Bochner technique to harmonic maps. We also announce our results on harmonic maps that we have obtained using the Bochner technique which are published in the present paper.The Bochner technique is the most important analytic method of differential geometry "in the large" which derived by Bochner for proving so-called vanishing theorems under appropriate curvature conditions on compact Riemannian manifolds (see [25];[32] and [33]). In the works of Lichnerowicz, Nomizu, Kodaira, Yano and others, the analytic Bochner method was substantially developed and successfully applied to complex, complete Riemannian, and Lorentzian manifolds. In particular, the Bochner technique has applications in the theory of harmonic maps "in the large" which presented in the monographs [13] and [18] from the present point of view. At the same time, we note that the reader is referred to [1, pp. 65-105]; [5]; [6] and [7] for a detailed account of harmonic maps. ____________________________________ g) and ( g M , ) is totally geodesic if the section curvature of ( g M , ) is non-negative and (M, g) is a compact manifold with the Ricci tensor Ric f Ric * ≥ for the pullback Ric f * of the Ricci tensor Ric by f. Beside it, in the fifth section we will show that a In the case of Kähler manifolds Sealey in [23] studied holomorphic maps (which are harmonic) between Kähler manifolds (M, g, J) and ( J , g , M ). Another generalization of the Bochner technique was presented by Siu in [24] where he has obtained a Weitzenböck formula involving only the curvature of the image manifold for harmonic maps between Kähler manifolds (M, g, J) and ( J , g , M ). This type of argument enabled him to study properties of harmonic mappings and to obtain rigidity results between compact Kähler manifolds with curvature conditions on the image manifold only. A similar analysis was carried out by Sampson in [20] for harmonic maps from a compact Kähler manifold (M, g, J) into a Riemannian manifold ( ) g M , .In addition, we state that the reader is referred to [13] and [18] for a detailed account of harmonic maps between Kähler manifolds and their generalizations.In the sixth section of our paper we prove that any holomorphic mapping ( ) () between a complete almost semi-Kählerian manifold (M, g,

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“…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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Evolution of contractions by mean curvature flow

Cited by 16 publications

References 12 publications

Self-Expanders of the Mean Curvature Flow

Self-Expanders of the Mean Curvature Flow

Vanishing theorems for harmonic mappings into non-negatively curved manifolds and their applications

Rigidity of the Hopf fibration

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