Remarks on $L^{p}$-vanishing results in geometric analysis

Pigola, Stefano; Veronelli, Giona

doi:10.48550/arxiv.1011.5413

Cited by 1 publication

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“…For example, the authors in [36] proved that any stable complete minimal hypersurface with finite total scalar curvature is a hyperplane. We refer the reader to [17,24,30,31,37,42,43] for recent progress on this topic and references therein. According to our technique, we may establish some Bernstein theorems for minimal submanifolds with finite total scalar curvature by assuming some boundary condition of a compact sublevel set of the extrinsic exhaustion distance function (see Corollary 5.3, Corollary 5.4 and Remark 5.4), instead of assuming the stable condition.…”

Section: Introductionmentioning

confidence: 99%

Monotonicity Formulae, Vanishing Theorems and Some Geometric Applications

Hao

2013

The Quarterly Journal of Mathematics

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Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. Kähler) manifolds poss some real (resp. complex ) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae under suitable growth conditions on the energy of the p-forms. As an application, we establish a monotonicity formula for the Ricci form of a Kähler manifold of constant scalar curvature and then get a growth condition to derive the Ricci flatness of the Kähler manifold. In particular, when the curvature does not change sign, the Kähler manifold is isometrically biholomorphic to C m . Another application is to deduce the monotonicity formulae for volumes of minimal submanifolds in some outer spaces with suitable exhaustion functions. In this way, we recapture the classical volume monotonicity formulae of minimal submanifolds in Euclidean spaces. We also apply the vanishing theorems to Bernstein type problem of submanifolds in Euclidean spaces with parallel mean curvature. In particular, we may obtain Bernstein type results for minimal submanifolds, especially for minimal real Kähler submanifolds under weaker conditions.

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Section: Introductionmentioning

confidence: 99%

Monotonicity Formulae, Vanishing Theorems and Some Geometric Applications

Hao

2013

The Quarterly Journal of Mathematics

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show abstract

Remarks on $L^{p}$-vanishing results in geometric analysis

Cited by 1 publication

References 20 publications

Monotonicity Formulae, Vanishing Theorems and Some Geometric Applications

Monotonicity Formulae, Vanishing Theorems and Some Geometric Applications

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