A Note About the Strong Maximum Principle on RCD Spaces

Gigli, Nicola; Rigoni, Chiara

doi:10.4153/cmb-2018-022-9

Cited by 12 publications

(7 citation statements)

References 28 publications

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“…Then there exists c ∈ R such that sup ∂Ω u < c < ess sup Ω , in particular from the upper semicontinuity and compactness of ∂Ω we have that f − min(f, c) ∈ W 1,2 (X) with compact support in Ω. From this point the proof continue exactly as in [GR19,Thm. 2.3].…”

Section: Localized Bochner Inequalitymentioning

confidence: 65%

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

Gigli¹,

Violo²

2021

Preprint

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We generalize to the RCD(0, N ) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with non-negative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting.Motivated by the recent work in [AFM] we also introduce the notion of electrostatic potential in RCD spaces, which also satisfies our monotonicity formulas.Our arguments are mainly based on new estimates for harmonic functions in RCD(K, N ) spaces and on a new functional version of the '(almost) outer volume come implies (almost) outer metric cone' theorem.

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Section: Localized Bochner Inequalitymentioning

confidence: 65%

“…For the second part we apply the strong-maximum principle in [GR19,Thm. 2.8] to a ball B r (x) ⊂ Ω, obtaining that u is constantly equal to sup Ω u in B r (x).…”

Section: Localized Bochner Inequalitymentioning

confidence: 99%

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

Gigli¹,

Violo²

2021

Preprint

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“…Therefore we reach a contradiction, since g would be a non constant superharmonic function attaining its minimum at an interior point, see [63,Theorem 2.8].…”

Section: The Distance Function From Isoperimetric Setsmentioning

confidence: 99%

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Antonelli¹,

Pasqualetto²,

Pozzetta³

et al. 2022

Preprint

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This paper studies sharp and rigid isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called N -dimensional RCD(K, N ) spaces (X, d, H N ). Thanks to these results, we determine the asymptotic isoperimetric behaviour for small volumes in great generality, and for large volumes when K = 0 under an additional noncollapsing assumption. Moreover, we obtain new stability results for isoperimetric regions along sequences of spaces with uniform lower Ricci curvature and lower volume bounds, almost regularity theorems formulated in terms of the isoperimetric profile, and enhanced consequences at the level of several functional inequalities.The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination with an asymptotic mass decomposition result of perimeter-minimizing sequences.Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies. Contents 1. Introduction Isoperimetry and lower Ricci curvature bounds Main results Consequences and strategies of the proofs Sharp asymptotic behaviour for small and large volumes Comparison with the previous literature Acknowledgements 2. Preliminaries 2.1. Convergence and stability results 2.2. BV functions and sets of finite perimeter in metric measure spaces 2.3. Sobolev functions, Laplacians and vector fields in metric measure spaces 2.4. Geometric Analysis on RCD spaces 2.5. Localization of the Curvature-Dimension condition 3. The distance function from isoperimetric sets 4. Concavity properties of the isoperimetric profile function and consequences 4.1. Sharp concavity inequalities for the isoperimetric profile 4.2. Fine properties of the isoperimetric profile 4.3. Consequences 5. Stability of isoperimetric sets

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“…➌ Taking advantage of the calculus of measured-valued Laplacian along with some Γ -calculus, we show W ➍ By means of the first variation formula (see [54]) along with some Γ -calculus computations and by using a Sobolev-to-Lipschitz property, we show α := ➎ Using the strong maximum principle due to [52] and since α (which has been proven to be Lipschitz in the previous steps) is subharmonic, X is compact and α achieves an interior maximum, we deduce α is identically equal to 1. This implies the weak harmonicity of f := sin −1 • u and indeed, the stronger fact that the contracted Hessian of f vanishes; it also implies the identity…”

Section: Theorem 12 Suppose Mmentioning

confidence: 99%

“…In the setting of RCD(K, N ) spaces, the strong maximum principle -in a straightforward manner -simplifies to the following statement which is the maximum principle we will resort to, later on in § 4. Theorem 2.26 (Strong maximum principle in RCD setting [52]). Let (X, d, m) be an RCD * (K, N ) space with 1 ≤ N ≤ ∞ and Ω ⊂ X an open and connected subset.…”

Section: First and Second Variation Formulae In Thementioning

confidence: 99%

The rigidity of sharp spectral gap in nonnegatively curved spaces

Ketterer¹,

Kitabeppu²,

Lakzian³

2021

Preprint

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We extend the celebrated rigidity of the sharp first spectral gap under Ric ≥ 0 to compact infinitesimally Hilbertian spaces with nonnegative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to so-called compact RCD(0, N ) spaces; this is a category of metric measure spaces which in particular includes (Ricci) nonnegatively curved Riemannian manifolds, Alexandorv spaces, Ricci limit spaces, Bakry-Émery manifolds along with products, certain quotients and measured Gromov-Hausdorff limits of such spaces. In precise terms, we show in such spaces, λ 1 = π 2 /diam 2 if and only if the space is one dimensional with a constant density function.

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A Note About the Strong Maximum Principle on RCD Spaces

Abstract: We give a quick and direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance.

Cited by 12 publications

References 28 publications

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

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

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

The rigidity of sharp spectral gap in nonnegatively curved spaces

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