Liouville properties

Colding, Tobias Holck; Minicozzi, William P.

doi:10.4310/iccm.2019.v7.n1.a10

Cited by 16 publications

(10 citation statements)

References 51 publications

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“…Instead a key here is a new localization inequality for the Gaussian L 2 norm. This new approach allows us to obtain the optimal dependence; see [CM13] for more. Similar localization ideas also play a role later in this paper.…”

Section: Introductionmentioning

confidence: 99%

Complexity of parabolic systems

Colding

Minicozzi

2020

Publ.math.IHES

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We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. We use this in a major application of our new methods to give the first general bounds on generic singularities of surfaces in arbitrary codimension.We also show sharp bounds for codimension in arguably some of the most important situations of general ancient flows. Namely, we prove that in any dimension and codimension any ancient flow that is cylindrical at −∞ must be a flow of hypersurfaces in a Euclidean subspace. This extends well-known classification results to higher codimension.The bound on the codimension in terms of the entropy is a special case of sharp bounds for spectral counting functions for shrinkers and, more generally, ancient flows. Shrinkers are solutions that evolve by scaling and are the singularity models for the flow.We show rigidity of cylinders as shrinkers in all dimension and all codimension in a very strong sense: Any shrinker, even in a large dimensional space, that is sufficiently close to a cylinder on a large enough, but compact, set is itself a cylinder. This is an important tool in the theory and is key for regularity; cf. [CM11].

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Section: Introductionmentioning

confidence: 99%

Complexity of parabolic systems

Colding

Minicozzi

2020

Publ.math.IHES

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“…2 Proposition 3.17. Assume that one of the following two conditions holds: 18. Suppose that {A z } z and {µ z } z are families of such periodic elliptic operators A z satisfying the same assumption of Theorem 3.13 and of rigged divisors µ z that depend continuously on a parameter z.…”

Section: Resultsmentioning

confidence: 99%

Liouville-Riemann-Roch theorems on abelian coverings

Kha¹,

Kuchment²

2019

Preprint

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Chapter 1. Introduction ix Chapter 2. Preliminaries 2.1. Periodic elliptic operators on abelian coverings 2.2. Floquet transform 2.3. Bloch and Fermi varieties 2.4. Floquet-Bloch functions and solutions 2.5. Liouville theorem on abelian coverings 2.6. Some properties of spaces V p N (A) 2.7. Explicit formulas for dimensions of spaces V ∞ N (A) 2.8. The Nadirashvili-Gromov-Shubin version of the Riemann-Roch theorem for elliptic operators on noncompact manifolds Chapter 3. The main results 3.1. Non-empty Fermi surface 3.2. Empty Fermi surface Chapter 4. Proofs of the main results 4.1. Proof of Theorem 3.6 4.2. Proof of Theorem 3.10 4.3. Proof of Theorem 3.13 4.4. Proof of Theorem 3.23 Chapter 5. Specific examples of Liouville-Riemann-Roch theorems 5.1. Self-adjoint operators 5.2. Non-self-adjoint second order elliptic operators Chapter 6. Auxiliary statements and proofs of technical lemmas 6.1. Properties of Floquet functions on abelian coverings 6.2. Basic properties of the family {A(k)} k∈C d 6.3. Properties of Floquet transforms on abelian coverings 6.4. A Schauder type estimate 6.5. A variant of Dedekind's lemma 6.6. Proofs of some other technical statements Chapter 7. Final Remarks and Acknowledgments 7.1. Remarks and conclusions 7.2. Acknowledgements Bibliography v

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“…Ancient solutions of polynomial growth were studied by many authors in Riemannian geometry, [Cal06,Cal07,LZ19,CM19]. Colding and Minicozzi [CM19] prove a theorem to compare the dimension of the space of ancient solutions of polynomial growth with that of the space of harmonic functions of polynomial growth on Riemannian manifolds. By following their argument, we extended the result to graphs.…”

Section: Definition 31 ([Bbi01]mentioning

confidence: 99%

Discrete harmonic functions on infinite penny graphs

Hua

2020

Preprint

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In this paper, we study discrete harmonic functions on infinite penny graphs. For an infinite penny graph with bounded facial degree, we prove that the volume doubling property and the Poincaré inequality hold, which yields the Harnack inequality for positive harmonic functions. Moreover, we prove that the space of polynomial growth harmonic functions, or ancient solutions of the heat equation, with bounded growth rate has finite dimensional property.

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Liouville properties

Cited by 16 publications

References 51 publications

Complexity of parabolic systems

Complexity of parabolic systems

Liouville-Riemann-Roch theorems on abelian coverings

Discrete harmonic functions on infinite penny graphs

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