scholarly journals Entropy, stability, and Yang–Mills flow

2016 ◽  
Vol 18 (02) ◽  
pp. 1550032 ◽  
Author(s):  
Casey Kelleher ◽  
Jeffrey Streets
Keyword(s):  
Linear Operator ◽  
Critical Points ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Entropy Stability ◽  
Yang Mills ◽  
Dimension Formula ◽  
Gap Theorem ◽  
Generic Singularities

Following [T. Colding and W. Minicozzi, II, Generic mean curvature flow I; generic singularities, Ann. of Math. 175(2) (2012) 755–833], we define a notion of entropy for connections over [Formula: see text] which has shrinking Yang–Mills solitons as critical points. As in [T. Colding and W. Minicozzi, II, Generic mean curvature flow I; generic singularities, Ann. of Math. 175(2) (2012) 755–833], this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stability in terms of the spectrum of a certain linear operator associated to the soliton. This leads furthermore to a gap theorem for solitons. These results point to a broader strategy of studying “generic singularities” of the Yang–Mills flow, and we discuss the differences in this strategy in dimension [Formula: see text] versus [Formula: see text].

scholarly journals Rigidity of generic singularities of mean curvature flow

2015 ◽  
Vol 121 (1) ◽  
pp. 363-382 ◽  
Author(s):  
Tobias Holck Colding ◽  
Tom Ilmanen ◽  
William P. Minicozzi
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Generic Singularities

scholarly journals Generic mean curvature flow I; generic singularities

2012 ◽  
Vol 175 (2) ◽  
pp. 755-833 ◽  
Author(s):  
Tobias Colding ◽  
William Minicozzi
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Generic Singularities

scholarly journals The singular set of mean curvature flow with generic singularities

2015 ◽  
Vol 204 (2) ◽  
pp. 443-471 ◽  
Author(s):  
Tobias Holck Colding ◽  
William P. Minicozzi
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Singular Set ◽  
Generic Singularities

scholarly journals ESTIMATES FOR EIGENVALUES OF $\mathfrak L$ OPERATOR ON SELF-SHRINKERS

2013 ◽  
Vol 15 (06) ◽  
pp. 1350011 ◽  
Author(s):  
QING-MING CHENG ◽  
YEJUAN PENG
Keyword(s):  
Differential Operator ◽  
Eigenvalue Problem ◽  
Mean Curvature ◽  
Stochastic Analysis ◽  
Mean Curvature Flow ◽  
Smooth Boundary ◽  
Curvature Flow ◽  
Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalue Problem ◽  
Generic Singularities

In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator [Formula: see text], which is introduced by Colding and Minicozzi in [Generic mean curvature flow I; generic singularities, Ann. Math.175 (2012) 755–833], on an n-dimensional compact self-shrinker in R n+p. Estimates for eigenvalues of the differential operator [Formula: see text] are obtained. Our estimates for eigenvalues of the differential operator [Formula: see text] are sharp. Furthermore, we also study the Dirichlet eigenvalue problem of the differential operator [Formula: see text] on a bounded domain with a piecewise smooth boundary in an n-dimensional complete self-shrinker in R n+p. For Euclidean space R n, the differential operator [Formula: see text] becomes the Ornstein–Uhlenbeck operator in stochastic analysis. Hence, we also give estimates for eigenvalues of the Ornstein–Uhlenbeck operator.

scholarly journals Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow

2020 ◽  
pp. 2050116
Author(s):  
R. Takahashi
Keyword(s):  
Line Bundle ◽  
Initial Data ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Phase Flow ◽  
Phase Operator ◽  
Yang Mills ◽  
Positive Time ◽  
Phase Operators

We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang–Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian phase flow (TLPF) on the space of almost calibrated [Formula: see text]-forms that fits into the GIT framework for dHYM metrics recently discovered by Collins–Yau. The TLPF has some special properties that are not seen for the line bundle mean curvature flow (i.e. the mirror of the Lagrangian mean curvature flow for graphs). We show that the TLPF starting from any initial data exists for all positive time. Moreover, we show that the TLPF converges smoothly to a dHYM metric assuming the existence of a [Formula: see text]-subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch.

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

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