arXiv research

The curvature-dimension condition implies a new weighted scalar curvature.

Paper proves Gromov's conjecture on manifolds with certain group properties.

The paper studies 3D manifolds with positive scalar curvature and volume growth.

The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.

New proof linking scalar curvature to volume growth on 3-manifolds.

Compactness theorem for manifolds with scalar curvature and entropy bounds.

Study on scalar curvature bounds and manifold topological complexity.

The report explores conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds.

Study shows tori metrics converging to flat under specific conditions.

The study proves that certain manifolds can have metrics with specific volume growth.

The paper proves compactness of warped product metrics on S²×S¹ with varying base metrics.

Paper bounds the A-hat genus using curvature and isoperimetric constants.

New examples show scalar curvature's role in sphere stability.

New method removes scalar curvature assumption in Ricci flow smoothing.

For sequences of warped product metrics on a $3$-torus satisfying the scalar curvature bound $R_j \geq -\frac{1}{j}$, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrin…

In an appendix to an earlier paper (cf. arXiv:1703.00984) we showed we showed how to construct tunnels of positive scalar curvature and of arbitrarily small length and volume connecting points in a \emph{three dimensional} manifold of \emph{constant sectional curvature}. Here we generalize the construction to arbitrary…

Paper studies convergence of nonnegative scalar curvature metrics to a specific limit space.

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 subco…

Preserves scalar curvature bounds under weak convergence of 3-manifolds.

We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension $n$, assuming uniform volume bounds and $L^{n/2}$ bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such bounds the underlying metric spaces are pre-compact in the Gr…

We show that non-collapsed Gromov-Hausdorff limits of polarized Kahler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson-Sun, in…

Paper proves convergence of warped product manifolds to a nonnegative scalar curvature limit.

Survey solves curvature problems with hyperbolic spaces.

Let g_t be a family of constant scalar curvature metrics on the total space of a Riemannian submersion obtained by shrinking the fibers of an original metric g, so that the submersion collapses as t approaches 0 (i.e., the total space converges to the base in the Gromov-Hausdorff sense). We prove that, under certain co…

The paper explores curvature-free effects in manifolds with volume growth and ends-counting.

Study compact sequences of warped product circles over spheres with nonnegative scalar curvature.

The paper constructs positive scalar curvature metrics via immersions.

Prove Gromov's Euclidean endpoint $C^0$ rigidity conjecture for positive mass theorem.

Proves a quantitative index theorem for positive scalar curvature metrics.

Survey on metrics and assembly maps in positive scalar curvature.

Paper proves Gromov's cube inequality in all dimensions.

Sharp inequalities for Kähler manifolds' systolic invariants are established.

The macroscopic version of Urysohn width for scalar curvature is disproven in high dimensions.

For any closed smooth Riemannian manifold H. Weyl has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of th…

Sharp volume growth ratio for 3D manifolds with positive scalar curvature.

We prove that given a hyperbolic manifold endowed with an auxiliary Riemannian metric whose sectional curvature is negative and whose volume is sufficiently small in comparison to the hyperbolic one, we can always find for any radius at least $1$ a ball in its universal cover whose volume is bigger than the hyperbolic …

Upper bound found for systolic geometry on manifolds with positive scalar curvature.

Study on 4-manifolds with positive scalar curvature.

3D manifold with positive curvature has a short geodesic.

Study crystallographic groups for positive scalar curvature conditions.

Paper answers Gromov's compactness question on noncompact manifolds.

New index theory proves Gromov's dihedral conjectures.

Proof of Gromov's theorem on convex polytopes with acute angles.

New examples of manifolds with positive scalar curvature and infinitely many poles.

The paper proves stability of positive mass theorem for flat 3-manifolds.

In this note we reprove a theorem of Gromov using Ricci flow. The theorem states that a, possibly non-constant, lower bound on the scalar curvature is stable under $C^0$-convergence of the metric.

The paper proves macroscopic versions of conjectures about scalar curvature and volume bounds.

The paper extends Gromov's K-cowaist to complete foliated manifolds and estimates leafwise scalar curvature.