Mean curvature flows and isotopy of maps between spheres

Tsui, Mao-Pei; Wang, Mu-Tao

doi:10.1002/cpa.20022

Cited by 53 publications

(43 citation statements)

References 23 publications

(52 reference statements)

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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: 79%

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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confidence: 79%

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

Степанов

Tsyganok

2017

manuscripta math.

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“…Now we have already shown that |λ i | ≤ 1−δ for some δ. By the theorem in [Tsui and Wang 2004] we know that this condition can be preserved along the mean curvature flow. After putting this into Equation (2-1), its right side term in parentheses becomes…”

Section: Long Time Existencementioning

confidence: 92%

Entire mean curvature flows of graphs

Han¹

2008

Pacific J. Math.

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“…Claim When t ≥ T 0 , ||B|| 2 ≤ c 19 e −τ t for some positive constants c 19 and τ . We prove the claim.…”

Section: Proofmentioning

confidence: 99%

Mean curvature flow of spacelike graphs

Salavessa

2010

Math. Z.

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We prove the mean curvature flow of a spacelike graph in ( 1 × 2 , g 1 − g 2 ) of a map f : 1 → 2 from a closed Riemannian manifold ( 1 , g 1 ) with Ricci 1 > 0 to a complete Riemannian manifold ( 2 , g 2 ) with bounded curvature tensor and derivatives, and with sectional curvatures satisfying K 2 ≤ K 1 , remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption K 2 ≤ K 1 , that if K 1 > 0, or if Ricci 1 > 0 and K 2 ≤ −c, c > 0 constant, any map f : 1 → 2 is trivially homotopic provided f * g 2 < ρg 1 where ρ = min 1 K 1 / sup 2 K + 2 ≥ 0, in case K 1 > 0, and ρ = +∞ in case K 2 ≤ 0. This largely extends some known results for K i constant and 2 compact, obtained using the Riemannian structure of 1 × 2 , and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one.

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Mean curvature flows and isotopy of maps between spheres

Cited by 53 publications

References 23 publications

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

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

Entire mean curvature flows of graphs

Mean curvature flow of spacelike graphs

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