Monotonicity formulas for harmonic functions in ${\rm RCD}(0,N)$ spaces

Gigli, Nicola; Violo, Ivan Yuri

doi:10.48550/arxiv.2101.03331

Cited by 5 publications

(7 citation statements)

References 26 publications

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“…Step 4. Let us complete the proof of (4.7) combining the previous ingredients with a limiting argument and [46,Proposition 5.3].…”

Section: Approximate Maximum Principles and Perturbation Argumentsmentioning

confidence: 97%

“…A key observation is that the Hopf-Lax semigroup plays a similar role of the exponential map, with the two-fold advantage of not needing smoothness of the ambient space and of communicating well with the synthetic lower Ricci bounds (thanks to a deep duality discovered by Kuwada [70], see also [6] for the extension to the RCD setting). The theory of Regular Lagrangian Flows [11] (see also [46] for some useful localised versions) is then a key tool in order to develop an Eulerian approach based on the continuity equation (well suited for the non-smooth RCD setting) of the smooth Lagrangian perspective given by classical Jacobi fields computation along geodesics.…”

Section: Approximate Maximum Principles and Perturbation Argumentsmentioning

confidence: 99%

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Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from $\mathrm{RCD}(K,N)$ spaces to $\mathrm{CAT}(0)$ spaces

Mondino¹,

Semola²

2022

Preprint

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We establish Lipschitz regularity of harmonic maps from RCD(K, N ) metric measure spaces with lower Ricci curvature bounds and dimension upper bounds in synthetic sense with values into CAT(0) metric spaces with non-positive sectional curvature. Under the same assumptions, we obtain a Bochner-Eells-Sampson inequality with a Hessian type-term. This gives a fairly complete generalization of the classical theory for smooth source and target spaces to their natural synthetic counterparts and an affirmative answer to a question raised several times in the recent literature. The proofs build on a new interpretation of the interplay between Optimal Transport and the Heat Flow on the source space and on an original perturbation argument in the spirit of the viscosity theory of PDEs.

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“…Step 4. Let us complete the proof of (4.7) combining the previous ingredients with a limiting argument and [46,Proposition 5.3].…”

Section: Approximate Maximum Principles and Perturbation Argumentsmentioning

confidence: 97%

Section: Approximate Maximum Principles and Perturbation Argumentsmentioning

confidence: 99%

Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from $\mathrm{RCD}(K,N)$ spaces to $\mathrm{CAT}(0)$ spaces

Mondino¹,

Semola²

2022

Preprint

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“…A related result bounding the number of ends of Alexandrov spaces with nonnegative sectional curvature outside a compact set was obtained in [8]. More recently, a result bounding the number of ends of RCD(0, N ) spaces was obtained in [7].…”

Section: Corollary Bmentioning

confidence: 97%

“…#F a ≤ 2 N (2 + t) N (1 − δ) −N t −N m.Adding up the contributions from the m families F a , we get that(7) k≤ 2 N (2 + t) N (1 − δ) −N t −N m 2 .…”

mentioning

confidence: 93%

On $\mathsf{CD}$ spaces with nonnegative curvature outside a compact set

Che,

Núñez-Zimbrón

2021

Preprint

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In this paper we adapt work of Z.-D. Liu to prove a ball covering property for non-branching CD spaces with nonnegative curvature outside a compact set. As a consequence we obtain uniform bounds on the number of ends of such spaces.Theorem 1.1. Let M n be a complete Riemannian manifold with nonnegative Ricci curvature outside a compact set B. Assume that Ric M ≥ (n − 1)H and that B ⊂ B D0 (p 0 ) for some p 0 ∈ M and D 0 > 0. Then for any µ > 0 there exists C = C(n, HD 2 0 , µ) > 0 such that for any r > 0, the following property is satisfied:We state and prove this result in the more general context of non-branching metric measure spaces satisfying the curvature-dimension condition introduced by Lott-Sturm-Villani [11,13,14] (see the section on preliminaries below for the definitions). This class of spaces contains the class of RCD spaces, as it was recently shown that these are non-branching (see [5, Theorem 1.3]), so, a fortiori, it also includes Alexandrov spaces [12,16] and weighted Riemannian manifolds. More precisely, we prove the following theorem.Theorem A. Let (X, d, m) be a non-branching metric measure space satisfying the CD(K, N ) condition for some N > 1 and K ∈ R. Assume that B is a compact subset of X with B ⊂ B D0 (p 0 ) for some p 0 ∈ M and D 0 > 0 and such that the CD loc (0, N ) condition is satisfied on X \ B (see Definition 2.5). Then for any µ > 0, there exists C = C(N, KD 2 0 , µ) > 0 such that for any r > 0, the following property is satisfied: If S ⊂ B r (p 0 ), there exist p 1 , . . . , p k ∈ S with k ≤ C and such thatThe proof follows the arguments of [9,10] almost verbatim, albeit with some needed adaptations to account for the more general hypotheses. The main tools we need are a version of the local-toglobal theorem for the CD condition (see Lemma 2.2) and a Bishop-Gromov inequality for certain

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“…Remark 4.5. Recently, N. Gigli and I. V. Violo [GV21] obtained the locally Hölder continuity of a solution to an obstructed problem on RCD(K, N )-spaces.…”

Section: Proof Fixed Any Ballmentioning

confidence: 99%

One-phase Free Boundary Problems on RCD Metric Measure Spaces

Chan¹,

Zhang²,

Zhu³

2021

Preprint

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In this paper, we consider a vector-valued one-phase Bernoullitype free boundary problem on a metric measure space (X, d, µ) with Riemannian curvature-dimension condition RCD(K, N ). We first prove the existence and the local Lipschitz regularity of the solutions, provided that the space X is non-collapsed, i.e. µ is the N -dimensional Hausdorff measure of X. And then we show that the free boundary of the solutions is an (N −1)-dimensional topological manifold away from a relatively closed subset of Hausdorff dimension N − 3. Contents 1. Introduction 1.1. The Bernoulli-type free boundary problems on Euclidean spaces 1.2. Free boundary problems in RCD-spaces and the main results 2. Preliminaries 2.1. RCD(K, N ) metric measure spaces and their calculus 2.2. Non-collapsed RCD(K, N ) metric measure spaces 2.3. Sets of finite perimeter and the reduced boundary 3. Existence of a minimizer 4. Hölder continuity of local minimizers 5. Lipschitz continuity of local minimizers 5.1. Mean value inequality 5.2. Lipschitz continuity of local minimizers of J Q 6. Local finiteness of perimeter for the free boundary 6.1. Nondegeneracy 6.2. Density estimates near the free boundary 6.3. Local finiteness of perimeter 7. Compactness and the Euler-Lagrange equation 8. Regularity of the free boundary Appendix A. Weiss-type monotonicity on cones References

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Monotonicity formulas for harmonic functions in ${\rm RCD}(0,N)$ spaces

Cited by 5 publications

References 26 publications

Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from $\mathrm{RCD}(K,N)$ spaces to $\mathrm{CAT}(0)$ spaces

Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from $\mathrm{RCD}(K,N)$ spaces to $\mathrm{CAT}(0)$ spaces

On $\mathsf{CD}$ spaces with nonnegative curvature outside a compact set

One-phase Free Boundary Problems on RCD Metric Measure Spaces

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