Christian Ketterer - Personal Webpage
Research Interests
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Geometric constructions
An important question is wether geometric constructions preserves synthetic curvature bounds.
For instance, metric cones and suspensions appear naturally in the context of metric spaces with lower and upper Alexandrov curvature bounds and preserve this type of curvature bounds.
I study cones and suspensions, and more generally warped products, in the context metric measure spaces with generalized lower Ricci curvature bounds. In [Ke13] I show that warped products preserve curvature bounds in the sense of Lott, Sturm and Villani provided the underlying metric measure spaces are weighted Finsler manifolds.
Using different methods in [Ke14] I prove that metric cones are RCD(0,N+1) if and only if the underlying metric measure space is RCD(N-1,N).
Future research projects concern doubling, gluing and self-gluing constructions for RCD spaces.
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Rigidity and almost rigidity theorems
Rigidity phenomena come along with geometric inequalities. A classical examples is the Riemannian maximal diameter theorem that characterizes the equality case in the Bonnet-Myers diameter estimate. More precisely, an n-dimensional Riemannian manifold with Ricci curvature bigger or equal than n-1 has diameter bounded by pi, and equality appears if and only if the Riemannian manifold is isometric to the standard sphere. In [Ke14] I generalize this result to RCD(N-1,N) spaces. I prove that equality in the generalized Bonnet-Myers theorem for RCD spaces appears if and only if the metric measure space is isomorphic to a spherical suspension. In [Ke14] I generalize Obtata's theorem for the Lichnerowicz spectral inequality of the Laplace operator to RCD(N-1,N) spaces where the equality case again only appears in the case of spherical suspensions. The Lichnerowicz inequality has a counterpart for infinite dimensional RCD spaces. In a joint work [GKKO17] with Nicola Gigli, Kazumasa Kuwada and Shin-Ichi Ohta we prove that equality in this case appears if and only if the space splits off a real line equipped with a Gaussian measure.
A question is if these rigidity results also appear for metric measure spaces with curvature in the sense of Lott, Sturm or Villani, or even for the so-called measure contraction property MCP. The MCP is much weaker than the condition RCD though it still can be used to characterize lower Ricci curvature bounds for Riemannian manifold. In [KR15] Tapio Rajala and I construct counterexamples for the above rigidity results if one only assumes the MCP.
In a recent paper [Ke19] I prove a Heintze-Karcher inequality for hypersurfaces in metric measure spaces. A notion of mean curvature is presented that I will further investigate, especially to obtain new rigidity phenomena.
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Sectional lower curvature via optimal transport
In [MK17] Andrea Mondino and I adress the question if the idea of characterizing lower Ricci curvature bounds via optimal transport is also applicable for other curvature quantities that arise from the curvature tensor of a Riemannian manifold. We consider so-called p-Ricci curvature that interpolates between the ususual Ricci tensor and sectional curvature. We show that p-Ricci curvature is bounded from below if and only if the relative entropy along Wasserstein geodesics, that for each time slice are supported on p-rectifiable sets, satifies a generalized convexity inequality.
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Variable Ricci lower curvature bounds
In [Ke15] I introduce a curvature-dimension condition CD(k,N) for metric measure spaces (X,d,m) where k is a variable lower bound of the curvature. The main result result is a generalization of the Bonnet-Myers estimate. More precisely, if k is strictly bounded from below by (N-1)/4 d(.,o)^2 for some point o in the space, then the space must be compact. The result is sharp since equality appears if the space is a warped product over the positive real line.
Our definition of variable curvature bounds allows to study family of spaces where the curvature bound is uniformily L^p-integrable or is contained in a Kato class. Especially stability questions are of interest.
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RCD spaces with upper curvature bounds
In collaboration with Vitali Kapovitch and Martin Kell we study RCD spaces that satisfy upper curvature bounds, either locally or globally. In the later case one says the space is CAT. In [KK18] Vitali Kapovitch and I first consider the noncollapsed case where we show that the condition RCD(K,n) together with an upper curvature bound k implies that the underlying metric space has Alexandrov curvature bounded from below by K-k(n-2). The exact value of the lower curvature bound is sharp as constant curvature space forms demonstrate. In the non-collapsed case this result fails, and in a joint work with Vitali Kapovitch and Martin Kell [KKK19] we study the fine structure of such spaces. We can show that general RCD spaces with upper curvature bounds are topological manifolds with boundary and their geometric interior is an open C^1 manifold equipped with a continuous BV Riemannian metric that induces the metric structure. The class of collapsed
RCD+CAT spaces is an interesting subclass that has better regularity properties than general RCD spaces, is still closed under measured Gromov-Hausdorff convergence and strictly larger than the class of Alexandrov spaces with curvature bounded below. The spaces in this class therefore can serve as a test case for more general conjectures.
Future research will adress topological consequences of the RCD+CAT condition, especially a generalisation of Gromov's almost flat manifold theorem.
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Stability of tori under lower scalar curvature bounds
During the summer school of the Fields program on geometric analysis in the summer 2017 together with Raquel Perales and Robin Neumeyer (later joined by Armando Cabrera Pacheco) I started a joint project on stability of the celebrated torus rigidity theorem under non-negative scalar curvature by Schoen-Yau and Gromov-Lawson. The later theorem states that a Riemannian torus with non-negative scalar curvature must be a flat torus. One can interpret this result as the scalar curvature counterpart of the Cheeger-Gromoll splitting theorem for Ricci curvature. A natural question to adress is in which sense is this result stable w.r.t. perturbation of the lower scalar curvature bound. A conjecture of Gromov precisely goes as follows: "There is a particular 'Sobolev type weak metric' in the space of n-manifolds X, such that, for example, (Riemannian) tori (X,g) with (scalar curvature) R_g\geq -\epsilon, when properly normalized, (sub)converge to flat tori for \epsilon going to 0, but these X may, in general, diverge in stronger metrics."
In [CKP] we solve this conjecture in the special case of graph tori and for the notion of intrinsic flat convergence that was introduced by Sormani and Wenger.
Future research concerns a solution of Gromov's conjecture for intrinsic flat convergence and a more general class of Riemannian tori.
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