14 problems

A simple Riemannian manifold is a compact Riemannian manifold with strictly convex boundary and no conjugate points, such that every pair of points is joined by a unique minimizing…

Burago–Ivanov conjecture. Every simple manifold is a strict minimal filling.

Boundary rigidity conjecture. Compact simple manifolds of any dimension are boundary rigid.

Boundary rigidity conjecture. The problem should be uniquely solvable for simple manifolds.

Let $\partial M$ be the boundary of a projective manifold, and suppose the non-rigidity condition holds, namely $i^*\alpha^0_\mu=0$ for all $\mu$, where…

Let $\overline{M}$ be an analytic manifold with boundary, endowed with an analytic projective structure, and let $\partial M$ be its boundary. Suppose that at every boundary point…

Let $M$ be a smooth manifold with boundary, and let $g_1$ and $g_2$ be smooth metrics of Anosov type on $M$ such that … Assume that $g_1$ and $g_2$ have the same marked lens data,…

Boundary rigidity conjecture. Boundary rigidity holds for compact simple Riemannian manifolds of any dimension.

Scattering rigidity conjecture. For simple manifolds, the scattering relation determines the isometry type of the manifold.

Let $\Omega \subset \mathbb{C}^d$ be a bounded convex domain and let $\xi_0\in\partial\Omega$. A holomorphic self-map is a holomorphic map $f:\Omega\to\Omega$. The tangent cone of…

Let $\Omega \subset \mathbb{C}^d$ be a pseudoconvex domain of finite type and let $\xi_0 \in \partial \Omega$. A holomorphic self-map is a map $f:\Omega\to\Omega$ that is holomorph…

Let $M$ be a compact Riemannian manifold with boundary. It is a strict minimal filling if, for every compact Riemannian manifold $M'$ with $\partial M'=\partial M$, the inequalitie…

Let $M$ be a simple compact Riemannian manifold with boundary, and let $bd_M$ denote its boundary distance function. A filling of $(\partial M,bd_M)$ is a compact orientable Rieman…

Boundary rigidity conjecture. Every simple manifold is boundary rigid: any other Riemannian manifold with boundary having the same boundary distance function is isometric to it.