Anton Petrunin

Semiconcave Functions in Alexandrov’s Geometry

The following is a compilation of some techniques in Alexandrovs geometry which are directly connected to convexity.

Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces

We show that almost nonnegatively curved m-dimensional manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.

Non-negative pinching, moduli spaces and bundles with infinitely many souls

We show that in each dimension

Asymptotical flatness and cone structure at infinity

n\ge 10

Harmonic functions on Alexandrov spaces and their applications

there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian

On intrinsic isometries to Euclidean space

n

An upper bound for the curvature integral

-manifolds with

A globalization for non-complete but geodesic spaces

0\le \sec\le 1

Monotonicity of saddle maps

, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non-homeomorphic. Finally, we construct examples of noncompact manifolds whose moduli spaces of complete metrics with

Smoothing 3-dimensional polyhedral spaces

\sec\ge 0