Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann He
- PDF / 950,563 Bytes
- 64 Pages / 547.087 x 737.008 pts Page_size
- 94 Downloads / 158 Views
GAFA Geometric And Functional Analysis
DISTRIBUTION-VALUED RICCI BOUNDS FOR METRIC MEASURE SPACES, SINGULAR TIME CHANGES, AND GRADIENT ESTIMATES FOR NEUMANN HEAT FLOWS Karl-Theodor Sturm Dedicated to the memory of Professor Kazumasa Kuwada Abstract. We will study metric measure spaces (X, d, m) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds BE1 (κ, ∞) • for which we prove the equivalence with sharp gradient estimates, • the class of which will be preserved under time changes with arbitrary ψ ∈ Lipb (X), and • which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets Y ⊂ X. In the latter case, the distribution-valued Ricci bound will be given by the signed measure κ = k mY + σ∂Y where k denotes a variable synthetic lower bound for the Ricci curvature of X and denotes a lower bound for the “curvature of the boundary” of Y , defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.
Contents 1 2
3 4
5
6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . W −1,∞ -valued Ricci bounds . . . . . . . . . . . . . . . . . . 2.1 Taming Semigroup. . . . . . . . . . . . . . . . . . . . . 2.2 Bochner Inequality BE1 (κ, ∞) and Gradient Estimate. Equivalence of BE2 (k, N ) and CD(k, N ) . . . . . . . . . . . Time-Change and Localization . . . . . . . . . . . . . . . . 4.1 Curvature-Dimension Condition under Time-Change. . 4.2 Localization. . . . . . . . . . . . . . . . . . . . . . . . 4.3 Singular Time Change. . . . . . . . . . . . . . . . . . . Gradient Flows and Convexification . . . . . . . . . . . . . 5.1 Gradient Flows for Locally Semiconvex Functions. . . 5.2 Convexification. . . . . . . . . . . . . . . . . . . . . . . 5.3 Bounds for the Curvature of the Boundary. . . . . . . Ricci Bounds for Neumann Laplacians . . . . . . . . . . . . 6.1 Neumann Laplacian and Time Change. . . . . . . . . 6.2 Time Re-Change. . . . . . . . . . . . . . . . . . . . . . 6.3 Boundary Measure and Boundary Local Time. . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
1649 1653 1653 1663 1667 1676 1676 1678 1679 1684 1684 1687 1690 1695 1695 1700 1702
GAFA
DISTRIBUTION-VALUED RICCI BOUNDS FOR METRIC MEASURE SPACES...
1649
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1709
1 Introduction Background Synthetic lower bounds for the Ricci curvature as introduced in the founda
Data Loading...
