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Christian Ketterer

Publications and source records attributed to Christian Ketterer.

At least 19 recordsLinked to original sources
arXiv2026

Convergence of Lorentzian spaces and curvature bounds for generalized cones

The goal of this article is twofold. We introduce a notion of convergence for Lorentzian pre-length spaces, $\ell$-convergence, that extends previous convergence notions in this context. We show that timelike curvature and timelike curvature-dimension bounds are stable under (measured) $\ell$-convergence. Then, we show that $\ell$-convergence is well adapted for generalized Lorentzian cones: a sequence of generalized cones $-I_i\times_{f_i}X_i$ converges in $\ell$ sense if the base $I_i$ and the fiber $X_i$ converge in GH sense and the functions $f_i$ converge uniformly. We use this to show sharp timelike curvature and timelike curvature-dimension bounds for such cones. Finally, we obtain a pre-compactness theorem for $\ell$-convergence in the class of smooth generalized cones that have a uniform lower bound on the full Ricci (or Riemann) curvature tensor.

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arXiv2025

Warped products over one-dimensional base spaces and the RCD condition

We prove the Riemannian curvature-dimension condition $\mathsf{RCD}(KN,N+1)$ for an $N$-warped product $B\times_f^N F$ over a one-dimensional base space $B$ with a Lipschitz function $f: B\rightarrow \mathbb R_{\geq 0}$, provided (1) $f$ is a $Kf$-concave function, (2) $f$ satisfies a sub-Neumann boundary condition $\frac{\partial f}{\partial n}\geq 0$ on $\partial B\backslash f^{-1}(0)$ and $F$ is a compact metric measure space satisfying (3) the condition $\mathsf{RCD}(K_F (N-1), N)$ with $K_F:= \sup_B \{ (Df)^2 + Kf^2\}$. The result is sharp, i.e. we show that (1), (2) and (3) are necessary for the validity of statement provided $K_F\geq 0$. In general, only a weaker statement is true. If $f$ is assumed to be $Kf$-affine, then the condition $\mathsf{RCD}(K N, N+1)$ for the $N$-warped product holds if and only if the condition $\mathsf{RCD}(K_F(N-1), N)$ holds for $F$ for any $K_F\in \mathbb R$.

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arXiv2025

Generalized cones admitting a curvature-dimension condition

We study (generalized) cones over metric spaces, both in Riemannian and Lorentzian signature. In particular, we establish synthetic lower Ricci curvature bounds \`a la Lott-Villani-Sturm and Ohta in the metric measure case, and \`a la Cavalletti-Mondino in Lorentzian signature. Here, a generalized cone is a warped product of a one-dimensional base space, which will be positive or negative definite, over a fiber that is a metric space. We prove that Riemannian or Lorentzian generalized cones over $\mathsf{CD}$-spaces satisfy the (timelike) measure contraction property $\mathsf{(T)MCP}$ - a weaker version of a (timelike) curvature-dimension condition $\mathsf{(T)CD}$. Conversely, if the generalized cone is a $\mathsf{(T)CD}$-space, then the fiber is a $\mathsf{CD}$-space with the appropriate bounds on Ricci curvature and dimension. In proving these results we develop a novel and powerful two-dimensional localization technique, which we expect to be interesting in its own right and useful in other circumstances. We conclude by giving several applications including synthetic singularity and splitting theorems for generalized cones. The final application is that we propose a new definition for lower curvature bounds for metric and metric measure spaces via lower curvature bounds for generalized cones over the given space.

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arXiv2025

Warped products and synthetic lower curvature bounds: an overview

This is a survey about the contruction of warped products between (semi-)Riemannian manifolds and metric (measure) spaces. The resulting spaces will be semi-Riemannian manifolds, metric (measure) spaces or Lorentzian metric and metric measure spaces. We present details of the contruction in each case and we will highlight important properties like fiber independence and the energy equation. Warped products behave nicely in relation with curvature lower bounds. Here we will focus on sectional and Ricci curvature lower bounds and their Lorentzian counterparts. Throughout the article we provide many examples and formulate questions and conjectures.

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arXiv2024

Glued spaces and lower curvature bounds

In this short note we survey theorems and provide conjectures on gluing constructions under lower curvature bounds in smooth and non-smooth context. Focusing on synthetic lower Ricci curvature bounds we consider Riemannian manifolds, weighted Riemannian manifolds, Alexandrov spaces, collapsed and non-collapsed $RCD$ spaces, and sub-Riemannian spaces.

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arXiv2023

Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds

We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov-Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov-Hausdorff closeness to a flat torus and an integral bound {on $r_M(x)$, the smallest eigenvalue of the Ricci tensor $\text{ric}_x$ in $x$}, imply the existence of a harmonic splitting map. Combining these results with Stern's inequality, we provide a new Gromov-Hausdorff stability theorem for flat $3$-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.

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arXiv2023

Glued spaces and lower Ricci curvature bounds

We consider Riemannian manifolds $M_i$, ${i=0,1}$, with boundary and $\Phi_i\in C^{\infty}(M_i)$ non-negative such that the pair $(M_i, \Phi_i)$ admits Bakry-Emery $N$-Ricci curvature bounded from below by $K$. Let $Y_0$ and $Y_1$ be isometric, compact components of the boundary of $M_0$ and $M_1$ respectively and assume $\Phi_0=\Phi_1$ on $Y_0\simeq Y_1$. We assume that $\Pi_0+\Pi_1=\Pi \geq 0$ (*), and $d\Phi_0(\nu_0)+ d\Phi_1(\nu_1)\leq \mbox{tr}\Pi$ on $Y_0\simeq Y_1$ (**) where $\Pi_i$ is the second fundamental form and $\nu_i$ is inner unit normal field along $\partial M_i$. We show that the metric glued space $M=M_0\cup_{\mathcal I}M_1$ together with the measure $\Phi d\mathcal H^n$ satisfies the curvature-dimension condition $CD(K,\lceil N \rceil)$ where $\Phi: M\rightarrow [0,\infty)$ arises tautologically from $\Phi_1$ and $\Phi_2$. Moreover, $(M, \Phi d\mathcal H^n)$ is the collapsed Gromov-Hausdorff limit of smooth, $\lceil N \rceil$-dimensional Riemannian manifolds with Ricci curvature bounded from below by $K- \epsilon$ and is also the measured Gromov-Hausdorff limit of smooth, weighted Riemannian manifolds such that the Bakry-Emery $\lceil N \rceil$-Ricci curvature is bounded from below by $K-\epsilon$. On the other hand we show that given a glued manifold as described it satisfies the curvature-dimension condition $CD(K,N)$ only if the condition (*) and (**) hold. The latter statement generalizes a theorem of Kosovski\u{\i} for sectional lower curvature bounds and especially applies for the unweighted case where a lower Ricci curvature bound and $\dim_{M_i}\leq N$ replaces a lower Bakry-Emery $N$-Ricci curvature bound.

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arXiv2023

The rigidity of sharp spectral gap in nonnegatively curved spaces

We extend the celebrated rigidity of the sharp first spectral gap under $Ric\ge0$ to compact infinitesimally Hilbertian spaces with non-negative (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) non-negatively curved Riemannian manifolds, Alexandrov 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, $λ= \frac{π^{2}}{\mathrm{diam}^2}$ if and only if the space is one dimensional with a constant density function. We use new techniques mixing Sobolev theory and singular $1D$-localization which might also be of independent interest. As a consequence of the rigidity in the singular setting, we also derive almost rigidity results.

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arXiv2023

Characterization of the null energy condition via displacement convexity of entropy

We characterize the null energy condition for an $(n+1)$-dimensional Lorentzian manifold in terms of convexity of the relative $(n-1)$-Renyi entropy along displacement interpolations on null hypersurfaces. More generally, we also consider Lorentzian manifolds with a smooth weight function and introduce the Bakry-Emery $N$-null energy condition that we characterize in terms of null displacement convexity of the relative $N$-Renyi entropy. As application we then revisit Hawking's area monotonicity theorem for a black hole horizon and the Penrose singularity theorem from the viewpoint of this characterization and in the context of weighted Lorentzian manifolds.

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arXiv2023

Rigidity of mean convex subsets in non-negatively curved RCD spaces and stability of mean curvature bounds

We prove splitting theorems for mean convex open subsets in RCD (Riemannian curvature-dimension) spaces that extend results by Kasue, Croke and Kleiner for Riemannian manifolds with boundary to a non-smooth setting. A corollary is for instance Frankel's theorem. Then, we prove that our notion of mean curvature bounded from below for the boundary of an open subset is stable w.r.t. to uniform convergence of the corresponding boundary distance function. We apply this to prove almost rigidity theorems for uniform domains whose boundary has a lower mean curvature bound.

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arXiv2020

Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

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arXiv2020

Stability of metric measure spaces with integral Ricci curvature bounds

In this article we study stability and compactness w.r.t. measured Gromov-Hausdorff convergence of smooth metric measure spaces with integral Ricci curvature bounds. More precisely, we prove that a sequence of $n$-dimensional Riemannian manifolds subconverges to a metric measure space that satisfies the curvature-dimension condition $CD(K,n)$ in the sense of Lott-Sturm-Villani provided the $L^p$-norm for $p>\frac{n}{2}$ of the part of the Ricci curvature that lies below $K$ converges to $0$. The results also hold for sequences of general smooth metric measure spaces $(M,g_M, e^{-f}\mbox{vol}_M)$ where Bakry-Emery curvature replaces Ricci curvature. Corollaries are a Brunn-Minkowski-type inequality, a Bonnet-Myers estimate and a statement on finiteness of the fundamental group. Together with a uniform noncollapsing condition the limit even satisfies the Riemannian curvature-dimension condition $RCD(K,N)$. This implies volume and diameter almost rigidity theorems.

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arXiv2020

Stability of graphical tori with almost nonnegative scalar curvature

By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to $0$. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of $3$-dimensional tori $M_j$ that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form $R_{g_{M_j}} \geq -1/j$. We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang-Lee, Huang-Lee-Sormani and Allen-Perales-Sormani. Furthermore, our results also hold when the condition on the scalar curvature of a torus $(M, g_M)$ is replaced by a bound on the quantity $-\int_T \min\{R_{g_M},0\} d{\mbox{vol}_{g_T}}$, where $M=\mbox{graph}(f)$, $f: T \to \mathbb R$ and $(T,g_T)$ is a flat torus. Using arguments developed by Alaee, McCormick and the first named author after this work was completed, our results hold for dimensions $n \geq 4$ as well.

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arXiv2020

On gluing Alexandrov spaces with lower Ricci curvature bounds

In this paper we prove that in the class of metric measure spaces with Alexandrov curvature bounded from below the Riemannian curvature-dimension condition $RCD(K,N)$ with $K\in \mathbb{R}$ and $N\in [1,\infty)$ is preserved under doubling and gluing constructions.

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arXiv2020

The Heintze-Karcher inequality for metric measure spaces

In this note we prove the Heintze-Karcher inequality in the context of essentially non-branching metric measure spaces satisfying a lower Ricci curvature bound in the sense of Lott-Sturm-Villani. The proof is based on the the needle decomposition technique for metric measure spaces introduced by Cavalletti-Mondino. Moreover, in the class of spaces satisfying a Riemannian curvature-dimension condition with positive curvature the equality case is characterized.

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arXiv2019

On the structure of RCD spaces with upper curvature bounds

We develop a structure theory for RCD spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is a topological manifold with boundary whose interior is equal to the set of regular points. Further the set of regular points is a smooth manifold and is geodesically convex. Around regular points there are DC coordinates and the distance is induced by a continuous BV Riemannian metric.

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arXiv2018

Sectional and intermediate Ricci curvature lower bounds via Optimal Transport

The goal of the paper is to give an optimal transport characterization of sectional curvature lower (and upper) bounds for smooth $n$-dimensional Riemannian manifolds. More generally we characterize, via optimal transport, lower bounds on the so called $p$-Ricci curvature which corresponds to taking the trace of the Riemann curvature tensor on $p$-dimensional planes, $1\leq p\leq n$. Such characterization roughly consists on a convexity condition of the $p$-Renyi entropy along $L^{2}$-Wasserstein geodesics, where the role of reference measure is played by the $p$-dimensional Hausdorff measure. As application we establish a new Brunn-Minkowski type inequality involving $p$-dimensional submanifolds and the $p$-dimensional Hausdorff measure.

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