We prove local rigidity of every smooth conformally Euclidean metric for the anisotropic Calder\'on problem on smooth compact domains $M\subset\mathbb R^n$, $n\ge3$. Any smooth Riemannian metric $g$ sufficiently close to a fixed background $g_0=e^{2c}\mathsf e$ in $H^s(M)$, with integer $s>n/2+1$, having the same Dirichlet-to-Neumann map satisfies $g=\Phi^*g_0$ for a smooth diffeomorphism $\Phi$ fixing the boundary pointwise.
Let $(M^n,g)$ be a complete connected noncompact smooth Riemannian manifold without boundary, with $n\ge3$ and positive scalar curvature. We prove that $M$ admits a complete smooth metric of uniformly positive scalar curvature if $$ \liminf_{d_g(o,x)\to\infty}d_g(o,x)^2R_g(x)>\frac{n-1}{n} $$ for a fixed point $o\in M$...
We prove rigidity near the Euclidean metric for the anisotropic Calder\'on problem on smooth bounded connected domains in dimensions three and higher. Any smooth Riemannian metric sufficiently close to the Euclidean metric in a suitable H\"older norm and having the same Dirichlet-to-Neumann map agrees with it up to a d...
Let $U$ be a connected open subset of the complex unit ball $B^n$, $n\geq2$, and let $F_i: U \to \Omega_i$ be non-constant holomorphic maps into irreducible bounded symmetric domains. We prove that if $F=(F_1, \cdots, F_m)$ is a holomorphic isometric map from $U$ to the product of $\Omega_i$, then every $F_i$ extends t...
Let $n\ge 2$, and let $M$ be either the $n$-dimensional sphere $\mathbb{S}^n$ or the real projective space $\mathbb{RP}^n$. Given $\alpha\ge 1$, we construct a finitely generated group $G_{n,\alpha}$ that embeds into the group of $C^\alpha$-diffeomorphisms of $M$, but does not embed into the group of $C^\beta$-diffeomo...
Maximiliano Escayola, Carl-Fredrik Nyberg-Brodda· 0 citations
We prove that a local minimal hypersurface $M^3$ in the unit sphere $S^4$ with constant scalar curvature and three distinct principal curvatures at every point must be locally congruent to the minimal Cartan isoparametric hypersurface in $S^4$, which provides a positive answer to the Bryant conjecture \cite{ChangLocal}...
For $n\ge2$, we prove a quantitative centre-rigidity estimate for bounded sets of finite Riemannian perimeter that are stationary and volume-constrained stable for normalized Riemannian perimeter with prescribed curvature $h_j$. Under confinement to a fixed compact set and cutoffs of vanishing full surface Dirichlet en...
