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We show that every closed nonpositively curved surface satisfies Loewner’s systolic inequality. The proof relies on a combination of the Gauss–Bonnet formula with an averaging argument using the invariance of the Liouville measure under the geodesic flow. This enables us to find a disk with large total curvature around its center yielding a large area. [ABSTRACT FROM AUTHOR]

This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on N-dimensional RCD (K , N) spaces (X , d , H N) . Moreover, we obtain almost regularity theorems formulated in terms of the isoperimetric profile and enhanced consequences at the level of several functional inequalities. Most of our statements seem to be new even in the classical setting of smooth, non compact manifolds with lower Ricci curvature bounds. The synthetic theory plays a key role via compactness and stability arguments. [ABSTRACT FROM AUTHOR]

We generalize McCann’s theorem of optimal transport to a submanifold setting and use it to prove Michael–Simon–Sobolev inequalities for submanifolds in manifolds with lower bounds on intermediate Ricci curvatures. The results include a variant of the sharp Michael–Simon–Sobolev inequality in Brendle’s () when the intermediate Ricci curvatures are nonnegative. [ABSTRACT FROM AUTHOR]

We combine the pointed Gromov-Hausdorff metric with the locally C 0 -distance to obtain the pointed C 0 -Gromov-Hausdorff distance between maps of possibly different non-compact pointed metric spaces. The latter is combined with Walters’s locally topological stability proposed by Lee–Nguyen–Yang, and GH-stability from Arbieto-Morales to obtain the notion of topologically GH-stable pointed homeomorphism. We give one example to show the difference between the distance when taking different base points in a pointed metric space. [ABSTRACT FROM AUTHOR]

In this paper, we first prove a Kazdan–Warner type identity for the problem of prescribing weighted scalar curvature. We then study weighted Yamabe solitons and prove some rigidity results. [ABSTRACT FROM AUTHOR]

The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m), the weighted Yamabe problem consists on finding another smooth metric measure space conformal to (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ+μe−ϕ∕m\lambda +\mu {e}^{-\phi /m} for some constants μ\mu and λ\lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m). On the other hand, we prove a version of the conformal Schwarz lemma on (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m). All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.

We extend to p-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise averaged mappings. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations. This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to simply verifying that the individual components have the required regularity pointwise at fixed points of the splitting algorithms. Our convergence analysis differs from what can be found in the previous literature in that the regularity assumptions are only with respect to fixed points. Indeed we show that, if the fixed point mapping is pointwise nonexpansive at all cluster points, then these cluster points are in fact fixed points, and convergence of the sequence follows. Additionally, we provide a quantitative convergence analysis built on the notion of gauge metric subregularity, which we show is necessary for quantifiable convergence estimates. This allows one for the first time to prove convergence of a tremendous variety of splitting algorithms in spaces with curvature bounded from above. [ABSTRACT FROM AUTHOR]

We prove the macroscopic cousins of three conjectures: (1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, (2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, (3) a conjectural bound of ℓ 2 -Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover. [ABSTRACT FROM AUTHOR]

In this paper we give a complete description of the horofunction boundary of the infinite dimensional real hyperbolic space, and characterise its Busemann points. [ABSTRACT FROM AUTHOR]

We introduce a new distance, a Lipschitz-Prokhorov distancedLP, on the set Pℳ of isomorphism classes of pairs (X, P) where X is a compact metric space and P is the law of a continuous stochastic process on X. We show that (Pℳ,dLP) is a complete metric space. For Markov processes on Riemannian manifolds, we study relative compactness and convergence. [ABSTRACT FROM AUTHOR]

The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O( g) shortest paths. [ABSTRACT FROM AUTHOR]

In their previous paper [7] , the authors introduced the straight (non-game-theoretic) counterpart of the finite decomposition complexity defined by E. Guentner, R. Tessera, G. Yu. In the present paper, we correct a proof of a statement from [7] that the straight finite decomposition complexity implies Property A. We also discuss a dimensional-like ordinal-valued invariant related to the straight finite decomposition complexity. [ABSTRACT FROM AUTHOR]

In this paper we develop a structure theory of Einstein manifolds or manifolds with lower Ricci curvature bounds for certain singular spaces that arise as geometric limits of sequences of Riemannian manifolds. This theory generalizes the results that were obtained by Cheeger, Colding and Naber in the smooth setting. In the course of the paper, we will carefully characterize the assumptions that we have to impose on this sequence of Riemannian manifolds in order to guarantee that the individual results hold. An important aspect of our approach is that we don't need impose any Ricci curvature bounds on the sequence of Riemannian manifolds leading to the singular limit. The Ricci curvature bounds will only be required to hold on the regular part of the limit and we will not impose any (synthetic) curvature condition on its singular part. The theory developed in this paper will have applications in the blowup analysis of certain geometric equations in which we study scales that are much larger than the local curvature scale. In particular, this theory will have applications in the study of Ricci flows of bounded scalar curvature, which we will describe in a subsequent paper. [ABSTRACT FROM AUTHOR]

In this article we prove comparative results on the Gromov hyperbolicity of plane domains equipped with the quasihyperbolic metric. By a comparative result we mean one which assumes hyperbolicity in one domain and obtains it in a different domain somehow related to the original one. We derive a characterization (simple to check in practical cases) of the Gromov hyperbolicity of a plane domain Ω* obtained by deleting from the original domain Ω any uniformly separated union of compact sets. We present as well a result about stability of hyperbolicity. [ABSTRACT FROM AUTHOR]