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Misha Gromov

Publications and source records attributed to Misha Gromov.

At least 19 recordsLinked to original sources
arXiv2024

Torsion Obstructions to Positive Scalar Curvature

We study obstructions to the existence of Riemannian metrics of positive scalar curvature on closed smooth manifolds arising from torsion classes in the integral homology of their fundamental groups. As an application, we construct new examples of manifolds which do not admit positive scalar curvature metrics, but whose Cartesian products admit such metrics.

math.DG↗
arXiv2024

Dynamical Morse entropy

We consider actions of a tileable amenable group $Γ$ on a topological space $X$. For a continuous function on $X$, we define the entropy of the number of homologically detectable critical point of the average of that function over $Γ$. This number is bounded below by the sum of the Betti number entropy. This result is thus a generalization of a standard Morse inequality in differential geometry to this setting.

math.DS↗
arXiv2023

Scalar Curvature, Injectivity Radius and Immersions with Small Second Fundamental Forms

We prove in special cases the following. $\bullet_{Sc}$ Bounds on the {\it injectivity radii} of "topologically complicated" Riemannian $n$-manifolds $X$, where the scalar curvatures of $X$ are bounded from below, $Sc(X)\geq σ>0$. $\bullet_{curv}$ Lower bounds on {\it focal radii} of smooth immersions from $k$-manifolds, e.g. homeomorphic to the $k$-torus, to certain Riemannian manifolds of dimensions $n=k+m$, e.g. to the cylinders $S^{n-1} \times \mathbb R^1$. $\bullet_{mean}$ Topological lower bounds on the mean curvatures of domains in Riemannian manifolds. e.g. in the Euclidean $n$-space $\mathbb R^n$. At the present moment, our results are limited by the {\it spin condition} and the {\it $n\leq 8$ restriction.}

math.DG↗
arXiv2023

Isometric Immersions with Controlled Curvatures

We $δ$-approximate strictly short (e.g. constant) maps between Riemannin manifolds $f_0:X^m\to Y^N$ for $N>>m^2/2$ by $C^\infty$-smooth isometric immersions $f_δ:X^m\to Y^N$ with curvatures $curv(f_δ) < \frac {\sqrt 3}δ$, for $δ\to 0$

math.DG↗
arXiv2023

Convex Polytopes, Dihedral Angles, Mean Curvature and Scalar Curvature

We approximate boundaries of convex polytopes by smooth hypersurfaces $Y=Y_\varepsilon$ with {\it positive mean curvatures} and, by using basic geometric relations between the scalar curvatures of Riemannin manifolds and the mean curvatures of their boundaries, establish {\it lower bound on the dihedral angles} of these polytopes.

math.DG↗
arXiv2021

Area and Gauss-Bonnet inequalities with scalar curvature

Let $X$ be an $n$-dimensional Riemannian manifold with "large positive" scalar curvature. In this paper, we prove in a variety of cases that if $X$ "spreads" in $(n-2)$ directions {\it "distance-wise"}, then it {\it can't} much "spread" in the remaining 2-directions {\it "area-wise".} Here is a geometrically transparent example of what we plan prove in this regard that illustrates the idea. Let $g$ be a Riemannin metric on $X= S^2\times \mathbb R^{n-2}$, for which the submanifolds $$\mbox {$\mathbb R_s^{n-2}=s\times \mathbb R^{n-2}\subset X$ and $S^2_y= S^2\times y \subset X$}$$ are {\it mutually orthogonal} at all intersection points $$x=(s,y)\in X=\mathbb R_s^{n-2}\cap S^2_y.$$ (An instance of this is $g=g(s,y)=ϕ(s,y)^2ds^2+ψ(s,y)^2dy^2$.) Let the Riemannian metric on $\mathbb R_s^{n-2}$ induced from $(X,g)$, that is $g|_{\mathbb R_s^{n-2}}$, be {\it greater than the Euclidean} metric on $\mathbb R_s^{n-2} =\mathbb R^{n-2}$ for all $s\in S^2$. (This is interpreted as "large spread" of $g$ in the $(n-2)$ Euclidean directions.) {\sf If the {\it scalar curvature of $g$ is strictly greater than that of the unit 2-sphere}, $$Sc(g) \geq Sc(S^2)+\varepsilon=2+\varepsilon, \mbox { }\varepsilon>0,$$ then, provided $n\leq 7$, } (this, most likely, is unnecessary) {\sf there exists a smooth {\it non-contractible} spherical surface $S\subset X$, such that $$area(S)<area(S^2)=4π.$$} (This says, in a way, that $(X,g)$ "doesn't spread much area-wise" in the 2 directions complementary to the Euclidean ones.)

math.DG↗
arXiv2021

Four Lectures on Scalar Curvature

We overview main topics and ideas in spaces with their scalar curvatures bounded from below, and present a more detailed exposition of several known and some new geometric constraints on Riemannian spaces implied by the lower bounds on their scalar curvatures

math.DG↗
arXiv2020

No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds

A metric space $X$ is called uniformly acyclic if there there exists an {\it acyclicty control function} $R=R(r)=R_X(r)\geq r $, $0\leq r <\infty$, such that the homology inclusion homomorphisms between the balls around all points $x\in X$, $$H_i(B_x(r))\to H_i(B_x(R))$$ vanish for all $i=1,2,\ldots$. We show that if a complete orientable $m$-dimensional manifold $\tilde X$ of dimension $m\leq 5$ admits a proper (infinity goes to infinity) distance decreasing map to a complete $m$-dimensional uniformly acyclic manifold, then the scalar curvature of $\tilde X$ can't be uniformly positive, $$\inf _{x\in \tilde X}Sc(X,x) \leq 0.$$ Since the universal coverings $\tilde X$ of compact aspherical manifolds $X$ are {\it uniformly acyclic}, (in fact, {\it uniformly contractible}), these $X$, admit no metrics with $Sc>0$ for $dim (X)\leq 5$. Our argument, that depends on {\it torical symmetrization} of {\it stable $μ$-bubbles}, is inspired by the recent paper by Otis Chodosh and Chao Li on non-existence of metrics with $Sc>0$ on aspherical 4-manifolds and is also influenced by the ideas of Jintian Zhu and Thomas Richard.

math.DG↗
arXiv2019

Mean Curvature in the Light of Scalar Curvature

We formulate several conjectures on mean convex domains in the Euclidean spaces, as well as in more general spaces with lower bonds on their scalar curvatures, and prove a few theorems motivating these conjectures.

math.DG↗
arXiv2018

Hilbert Volume in Metric Spaces

We introduce a notion of Hilbertian n-volume in metric spaces with Besicovitch-type inequalities built-in into the definitions. which, ultimately, may turn useful for an approach to singular spaces with positive scalar curvature

math.MG↗
arXiv2018

Plateau Stein Manifolds

We study/construct (proper and non-proper) Morse functions on complete Riemannian manifolds, the level hypersurfaces of which have positive mean curvatures at all non-critical points. We show, for instance, that if a complete Rieannin manifold admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.

math.DG↗
arXiv2018

Metric Inequalities with Scalar Curvature

We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below, in the spirit of the classical bound on the distances between conjugates points in surfaces with positive sectional curvature.

math.DG↗