Stability of stationary maps of a functional related to pullbacks of metrics

Kawai, Shigeo; Nakauchi, Nobumitsu

doi:10.1016/j.difgeo.2015.11.005

Cited by 13 publications

(3 citation statements)

References 9 publications

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“…The method ( [51]) can be carried over to more general settings: Han and Wei find Φ-SSU manifolds (or Φ (2) -SSU manifolds) [24] and prove that every compact Φ-SSU manifold must be Φ-strongly unstable (Φ-SU), or Φ (2) -strongly unstable (Φ (2) -SU), i.e., every compact Φ-SSU manifold can neither be the domain nor the target of any nonconstant smooth stable Φ-harmonic map between two compact Riemannian manifolds and the homotopic class of maps (between two compact Riemannian manifolds) from or into Φ-SSU manifold contains a map of arbitrarily small Φ-energy. These generalize the cases S m , for m > 5, and compact minimal submanifolds of S m+p with Ric g > 3 4 mg, due to Kawai and Nakauchi [32,33], in which the nonexistence results of nonconstant stable Φ-harmonic maps are extended to Φ-SU or Φ (2) -SU features in [24]. Cases of SSU manifolds or Φ (1) -SSU manifolds such as spheres, product of spheres with appropriate dimensions, etc.…”

Section: Introductionmentioning

confidence: 87%

The Geometry of $$\Phi _S$$-Harmonic Maps

et al. 2021

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In this paper, we motivate and extend the study of harmonic maps or Φ (1) -harmonic maps (cf [15], Remark 1.3 (iii)), Φ-harmonic maps or Φ (2) -harmonic maps (cf. [24], Remark 1.3 (v)), and explore geometric properties of Φ (3) -harmonic maps by unified geometric analytic methods. We define the notion of Φ (3) -harmonic maps and obtain the first variation formula and the second variation formula of the Φ (3) -energy functional E Φ (3) . By using a stress-energy tensor and the asymptotic assumption of maps at infinity, we prove Liouville type results for Φ (3) -harmonic maps. We introduce the notion of Φ (3) -Superstrongly Unstable (Φ (3) -SSU) manifold and provide many interesting examples. By using an extrinsic average variational method in the calculus of variations (cf. [51,49]), we find Φ (3) -SSU manifold and prove that any stable Φ (3) -harmonic maps from a compact Φ (3) -SSU manifold (into any compact Riemannian manifold) or (from any compact Riemannian manifold) into a compact Φ (3) -SSU manifold must be constant. We also prove that the homotopy class of any map from a compact Φ (3) -SSU manifold (into any compact Riemannian manifold) or (from any compact Riemannian manifold) into a compact Φ (3) -SSU manifold contains elements of arbitrarily small Φ (3) -energy. We call a compact Riemannian manifold M to be Φ (3) -strongly unstable (Φ (3) -SU) if it is not the target or domain of a nonconstant stable Φ (3) -harmonic map (from or into any compact Riemannian manifold) and also the homotopy class of any map to or from M (from or into any compact Riemannian manifold) contains elements of arbitrarily small Φ (3) -energy. We prove that every compact Φ (3) -SSU manifold is Φ (3) -SU. As consequences, we obtain topological vanishing theorems and sphere theorems by employing a Φ (3) -harmoic map as a catalyst. This is in contrast to the approaches of utilizing a geodesic ([45]), minimal surface, stable rectifiable current ([34, 29, 50]), p-harmonic map (cf. [53]), etc., as catalysts. These mysterious phenomena are analogs of harmonic maps or Φ (1) -harmonic maps, p-harmonic maps, Φ S -harmonic maps, Φ S,p -harmonic maps, Φ (2) -harmonic maps, etc., (cf.

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Section: Introductionmentioning

confidence: 87%

The Geometry of $$\Phi _S$$-Harmonic Maps

et al. 2021

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“…The case N as in Corollary 5.4 satisfies Theorem 1.1 (a) and (c) are due to S. Kawai and N. Nakauchi (cf. [18]).…”

Section: Examples Of φ-Ssu Manifoldsmentioning

confidence: 99%

“…A Riemannian n-manifold N is said to be Φ-supersrongly unstable (Φ-SSU) if there exists an isometric immersion of N in R q with its second fundamental form B such that, for all unit tangent vectors x to N at every point y ∈ N, the following functional is always negativevalued: The cases (1) N is S n , n ≥ 5 and (2) N is a minimal submanifold in the unit sphere with Ric N ≥ 3 4 n satisfying properties (a) and (c) are due to S. Kawai and N. Nakauchi (cf. [17,18]). These are analogs of the following: S n , n > 2 is not the domain of any nonconstant stable harmonic maps into any Riemannian manifold due to Xin ([40]), S n , n > 2 is not the target of any nonconstant stable harmonic maps from any Riemannian manifold due to Leung [22] and Wei [34], and a minimal k-submanifold N in the unit sphere with Ric N > (1 − 1 p )k, p < k is neither the domain nor the target of any nonconstant stable pharmonic maps (cf.…”

Section: Introductionmentioning

confidence: 99%

$Φ$-Harmonic Maps and $Φ$-Superstrongly Unstable Manifolds

Han¹,

Wei²

2019

Preprint

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In this paper, we motivate and define Φ-energy density, Φ-energy, Φ-harmonic maps and stable Φ-harmonic maps. Whereas harmonic maps or p-harmonic maps can be viewed as critical points of the integral of σ 1 of a pull-back tensor, Φ-harmonic maps can be viewed as critical points of the integral of σ 2 of a pull-back tensor. By an extrinsic average variational method in the calculus of variations (cf. [13, 38, 37, 14]), we derive the average second variation formulas for Φ-energy functional, express them in orthogonal notation in terms of the differential matrix, and find Φ-superstrongly unstable (Φ-SSU) manifolds. We prove, in particular that every compact Φ-SSU manifold must be Φ-strongly unstable (Φ-SU), i.e., (a) A compact Φ-SSU manifold cannot be the target of any nonconstant stable Φ-harmonic maps from any manifold, (b) The homotopic class of any map from any manifold into a compact Φ-SSU manifold contains elements of arbitrarily small Φ-energy, (c) A compact Φ-SSU manifold cannot be the domain of any nonconstant stable Φ-harmonic map into any manifold, and (d) The homotopic class of any map from a compact Φ-SSU manifold into any manifold contains elements of arbitrarily small Φ-energy (cf. Theorem 1.1(a), (b), (c), and (d).) We also provide many examples of Φ-SSU manifolds, and establish a link of Φ-SSU manifold to p-SSU manifold and topology. The extrinsic average variational method in the calculus of variations that we have employed is in contrast to an average method in PDE that we applied in [5] to obtain sharp growth estimates for warping functions in multiply warped product manifolds.

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A Variation Problem for Stress–Energy Tensor

Han

2019

Results Math

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Stability of stationary maps of a functional related to pullbacks of metrics

Cited by 13 publications

References 9 publications

The Geometry of $$\Phi _S$$-Harmonic Maps

The Geometry of $$\Phi _S$$-Harmonic Maps

$Φ$-Harmonic Maps and $Φ$-Superstrongly Unstable Manifolds

A Variation Problem for Stress–Energy Tensor

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