We prove that every rational proper holomorphic map from $\mathbb{B}^2$ to $\mathbb{B}^4$ has degree at most $5$, confirming D'Angelo's conjecture in this case. This sharp bound follows from Meylan's degree estimates and an intersection-theoretic argument ruling out degree $6$.
Let $f:\mathbf P^n_{\mathbb C}\to\mathbf P^n_{\mathbb C}$ be an endomorphism of degree greater than one. We prove that every irreducible Zariski closed subset $V$ with $f^{-1}(V)=V$ is a linear subspace.
Let $M^n\subset\mathbb{S}^{n+1}(1)$ be a closed minimal hypersurface with constant $S=|h|^2$ and constant $f_3=\operatorname{tr} h^3$, where $h$ is the shape operator. We prove that $S>n$ implies $S\geq2n$, and that the equality images are precisely the minimal Cartan isoparametric hypersurfaces with three principal cu...
Jian-Quan Ge, Hui-Xin Tan, Wen-Jiao Yan et al.· 3 citations
Let $q \equiv 1 \bmod 8$ be a prime power. We prove that $L\left(\tfrac{1}{2}, \chi\right) \neq 0$ for 100% of the imaginary quadratic Dirichlet characters of $\mathbb{F}_q(T)$.
Peter Koymans, Carlo Pagano, M. Shusterman· 0 citations
In this work, we establish the sharp $L^2 \to L^4$ extension inequality for a large class of quadratic surfaces in $\mathbb{F}^3$, where $\mathbb{F}$ is a finite field of odd characteristic. These surfaces include the cones without the origin and the spheres with nonzero radius $j$ such that $-j$ is not a square in $\m...
Let $\mu$ be a finitely supported probability measure on $\mathrm{SL}_2(\mathbb{R})$ whose support generates a Zariski-dense discrete subgroup. We prove that its stationary probability measure on $\mathbb{P}^1(\mathbb{R})$ is singular with respect to the $\mathrm{SO}(2)$-invariant probability measure. The proof compare...
We construct smooth time-periodic Hamiltonians on the monotone manifolds $S^2\times S^2$ and $\mathbb{C}P^2$ for which the Clifford torus meets its Hamiltonian image in exactly two isolated degenerate points. The product example disproves the degenerate Arnol'd-Givental conjecture.
Wen-Min Gong· 0 citations
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