We prove a multidimensional Birkhoff theorem for a new class of"$C^0$ Lagrangian subsets"in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset $L$ under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then $L$ and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in $C^0$ symplectic topology.
We give a proof of a folklore conjecture that a compact K\"ahler manifold with the same scalar Laplace spectrum as complex projective space equipped with the Fubini--Study metric is holomorphically isometric to it. More generally, we prove that the spectrum of the Hodge Laplacian in any one fixed degree $p$ determines...
We extend the notion of non-local almost minimal sets introduced in [Dipierro, Valdinoci, and Villa, J. Math. Pures Appl. (2026)] by admitting lower-order powers for the remainder term. We combine classical techniques and a new, weaker version of the Euler-Lagrange inequalities (at a microscopic scale) to obtain $\math...
S. Dipierro, O. Savin, Enrico Valdinoci et al.· 0 citations
We introduce a notion of quasiconvexity for integrands defined on the bundle of linear maps between the tangent spaces of two smooth, connected and complete Riemannian manifolds $(M,g)$ and $(N,h)$, naturally extending the classical Euclidean notion. For continuous integrands, we prove that this Riemannian quasiconvexi...
Aurora Corbisiero, Chiara Leone, Carlo Mantegazza· 0 citations
Let $F:M^n\to\mathbb{S}^{2n+1}$ be a minimal Legendrian immersion of a connected closed manifold, where $n\ge3$, and let $B$ be its second fundamental form. We prove that if $|B|^2\le(n-1)(n+2)/n$, then, up to a unitary transformation, $F$ parametrizes a totally geodesic sphere or is a finite covering of the image of t...
We study global smooth Calabi flows on polarized K\"ahler surfaces $(M,L)$, assuming uniformly bounded Sobolev constants and the algebraic conditions $\chi(\mathcal O_M)=1$ and $c_1(M)\cdot c_1(L)>0$. Using the orbifold compactness theorem, we prove a uniform Bergman estimate, construct a normal test configuration for...
In one complex variable the Schwarzian derivative characterizes the M\"obius transformations, arises from a second-order linear equation, and controls univalence through the theorems of Nehari and of Ahlfors--Weill. This chapter presents what is known about these three properties for locally biholomorphic mappings in $...
R. Hernández· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
