Preprints - IAM Publications


	2019 Matthias Erbar and Karl-Theodor Sturm Riggidity of cones with bounded curvature to appear in JEMS, arxiv e-print 1712.08093 2019 https://arxiv.org/abs/1712.08093

	2018 Eva Kopfer and Karl-Theodor Sturm Heat flow on time-dependent metric measure spaces and super-Ricci flows Comm. Pure Appl. Math., 71(12): 2500--2608 2018 10.1002/cpa.21766

	Janna Lierl and Karl-Theodor Sturm Neumann heat flow and gradient flow for the entropy on non-convex domains Calc. Var. Partial Differential Equations, 57(1): Art. 25, 22 2018 10.1007/s00526-017-1292-8

	Angelo Profeta and Karl-Theodor Sturm Heat ow with Dirichlet boundary conditions via optimal transport and gluing of mm spaces arxiv e-print 1809.00936 2018 https://arxiv.org/abs/1809.00936

	Karl-Theodor Sturm Super-Ricci flows for metric measure spaces J. Funct. Anal., 275(12): 3504--3569 2018 10.1016/j.jfa.2018.07.014

	2017 Karl-Theodor Sturm Remarks about synthetic upper Ricci bounds for mm spaces arxiv e-print 1711.01707 2017 https://arxiv.org/abs/1711.01707

	2015 Matthias Erbar, Kazumasa Kuwada and Karl-Theodor Sturm On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces Invent. Math., 201(3): 993-1071 2015 http://dx.doi.org/10.1007/s00222-014-0563-7

	Nicola Gigli, Tapio Rajala and Karl-Theodor Sturm Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below J. Geom. Anal. 2015 http://arxiv.org/abs/1305.4849 Abstract: We prove existence and uniqueness of optimal maps on RCD∗(K,N) spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation and to the local-to-global property of RCD∗(K,N) bounds.

	Karl-Theodor Sturm Metric Measure Spaces with Variable Ricci Bounds and Couplings of Brownian Motions In Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda, Toshihiro Uemura, editor, Festschrift Masatoshi Fukushima, Volume 17 of Interdisciplinary Mathematical Sciences Chapter 27, page 553-575. 2015 http://dx.doi.org/10.1142/9789814596534_0027 Abstract: The goal of this paper is twofold: we study metric measure spaces (X, d, m) with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function k : X → ℝ we introduce the curvature-dimension condition CD(k, ∞) which canonically extends the curvature-dimension condition CD(K, ∞) of Lott-Sturm-Villani for constant K ∈ R. For infinitesimally Hilbertian spaces we prove its equivalence to an evolution-variation inequality EVIk which in turn extends the EVIK-inequality of Ambrosio-Gigli-Savaré; its stability under convergence and its local-to-global property. For metric measure spaces with uniform lower curvature bounds K we prove that for each pair of initial distributions µ1, µ2 on X there exists a coupling , t ≥ 0, of two Brownian motions on X with the given initial distributions such that a.s.

	2014 Shin-ichi Ohta and Karl-Theodor Sturm Bochner-Weitzenböck formula and Li-Yau estimates on Finsler manifolds Adv. Math., 252: 429-448 2014 http://dx.doi.org/10.1016/j.aim.2013.10.018

	Tapio Rajala and Karl-Theodor Sturm Non-branching geodesics and optimal maps in strong CD (K,$\backslash$ infty)-spaces Calculus of Variations and Partial Differential Equations, 50(3-4): 831--846 2014 http://dx.doi.org/10.1007/s00526-013-0657-x

	Karl-Theodor Sturm Gradient Flows for Semiconvex Functions on Metric Measure Spaces - Existence, Uniqueness and Lipschitz Continuity ArXiv e-prints 2014 http://arxiv.org/abs/1410.3966

	Karl-Theodor Sturm Ricci Tensor for Diffusion Operators and Curvature-Dimension Inequalities under Conformal Transformations and Time Changes ArXiv e-prints 2014 http://arxiv.org/abs/1401.0687 Abstract: Within the Γ2-calculus of Bakry and Ledoux, we define the Ricci tensor induced by a diffusion operator, we deduce precise formulas for its behavior under drift transformation, time change and conformal transformation, and we derive new transformation results for the curvature-dimension conditions of Bakry-Emery as well as for those of Lott-Sturm-Villani. Our results are based on new identities and sharp estimates for the N-Ricci tensor and for the Hessian. In particular, we obtain Bochner's formula in the general setting.

	Karl-Theodor Sturm A Monotone Approximation to the Wasserstein Diffusion In M. Griebel, editor, Singular Phenomena and Scaling in Mathematical Models, Volume 1 Chapter 2, page 25-48. Publisher: Springer International 2014 http://dx.doi.org/10.1007/978-3-319-00786-1_2

	2013 Martin Huesmann and Karl-Theodor Sturm Optimal transport from Lebesgue to Poisson The Annals of Probability, 41(4): 2426-2478 2013 http://dx.doi.org/10.1214/12-AOP814 Abstract: This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite. Moreover, we give precise conditions for the latter which demonstrate a sharp threshold at d=2d=2. The cost will be defined in terms of an arbitrary increasing function of the distance. The coupling will be realized by means of a transport map (“allocation map”) which assigns to each Poisson point a set (“cell”) of Lebesgue measure 1. In the case of quadratic costs, all these cells will be convex polytopes.

	Kazumasa Kuwada and Karl-Theodor Sturm Monotonicity of time-dependent transportation costs and coupling by reflection Potential Analysis, 39(3): 231-263 2013 http://dx.doi.org/10.1007/s11118-012-9327-4 Abstract: Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Émery’s curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distributions on spaceforms. As a corollary, we obtain a comparison theorem for the total variation between heat distributions. In addition, we show that our monotonicity is stable under the Gromov-Hausdorff convergence of the underlying space under a uniform curvature-dimension and diameter bound.

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