Related papers: Torus cannot collapse to a segment

In this paper, we show that every collapsed Gromov--Hausdorff limit of compact Heisenberg manifolds is isometric to a flat torus. Here we say that a sequence of sub-Riemannian manifolds collapses if their total measure with respect to the…

We consider noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature. We show that such solitons always dimension reduce at infinity. This generalizes an earlier result in [CDM22] to higher dimensions. In dimension…

Restrictions are obtained on the topology of a compact divergence-free null hypersurface in a four-dimensional Lorentzian manifold whose Ricci tensor is zero or satisfies some weaker conditions. This is done by showing that each null…

In this paper we describe the topology of 4-dimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature and with a uniform upper bound of diameter which collapse to metric spaces of lower dimensions.…

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus. A general theorem describing…

We analyze spectral minimal $k$-partitions for the torus. In continuation with what we have obtained for thin annuli or thin strips on a cylinder (Neumann case), we get similar results for anisotropic tori.

We study the spherical gravitational collapse of a compact object under the approximation that the radial pressure is identically zero, and the tangential pressure is related to the density by a linear equation of state. It turns out that…

Misner space is generalized to have the nonorientable topology of a Klein bottle, and it is shown that in a classical spacetime with multiply connected space slices having such a topology, closed timelike curves are formed. Different…

Given a hyperelliptic Klein surface, we construct companion Klein bottles. Bavard's short loops on companion bottles are studied in relation to the surface to improve an inequality of Gromov's in systolic geometry.

The structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus is studied. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the…

We give a short proof of the systolic inequality for the n-dimensional torus. The proof uses minimal hypersurfaces. It is based on the Schoen-Yau proof that an n-dimensional torus admits no metric of positive scalar curvature.

We prove the existence of a minimal (all leaves dense) foliation of codimension one, on every closed manifold of dimension at least 4 whose Euler characteristic is null, in every homotopy class of hyperplanes distributions, in every…

Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a…

The percolation threshold and wrapping probability $R_{\infty}$ for the two-dimensional problem of continuum percolation on the surface of a Klein bottle have been calculated by the Monte Carlo method with the Newman--Ziff algorithm for…

Let $(M^n_i, g_i)\to (X,d_X)$ be a Gromov-Hausdorff converging sequence of Riemannian manifolds with ${\rm Sec}_{g_i} \ge -1$, ${\rm diam}\, (M_i)\le D$, and such that the $M^n_i$ are all homeomorphic to tori $T^n$. Then $X$ is homeomorphic…

We consider a "Scalar-Einstein-Gauss-Bonnet" theory in four dimension, where the scalar field couples non minimally with the Gauss-Bonnet (GB) term. This coupling with the scalar field ensures the non topological character of the GB term.…

A compactness theorem is proved for a family of K\"{a}hler surfaces with constant scalar curvature and volume bounded from below, diameter bounded from above, Ricci curvature bounded and the signature bounded from below. Furthermore, a…

Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an…

In this note, we prove that any non-collapsing and compact Gromov-Hausdorff limit of Kahler-Einstein manifolds is either smooth or is orbifold outside a subvariety of complex codimension at least 3.

We consider stable minimal surfaces of genus 1 in Euclidean space and in Riemannian manifolds. Under the condition of covering stability (all finite covers are stable) we show that a genus 1 finite total curvature minimal surface in…