We prove local rigidity of the Euclidean metric and close conformally Euclidean ones for the anisotropic Calder\'on problem on smooth compact domains $M\subset\mathbb{R}^n$, $n\ge3$. A smooth Riemannian metric $g$ with the same induced boundary metric and Dirichlet-to-Neumann map as the background $g_0$ is isometric to $g_0$ by a diffeomorphism fixing the boundary pointwise if $\|g-g_0\|_{H^s(M)}$ is sufficiently small, where $s>n/2+1$ is an integer. For $g_0=e^{2c}\e$, we assume that $\|c\|_{C^k(M)}$ is sufficiently small, with an integer $k\ge s+2$ and $k>3n/2+5$.
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...
We study holomorphic isometric maps $F=(F_1,\cdots,F_m)$ from the complex unit ball $\mathbb{B}^n$, $n\geq2$, to a product of irreducible bounded symmetric domains $\Omega_1\times\cdots\times\Omega_m,$ equipped with positive constant multiples of their canonical K\"ahler--Einstein metrics. We prove that every nonconsta...
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}...
Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the vo...
