2026· Известия Российской академии наук Серия математическая· Vol 90, pp. 51-70· 0 citations· 21 references
Abstract
We show that if an upper semi-continuous function $\varphi$ on $\Omega\in\mathbb C^n$
satisfies some natural conditions, and for any non-negative psh function $\psi$ on
$\mathbb D\times]Omega$, the fiberwise Bergman kernel associated to $p_2^*\varphi+\psi$ is log-psh, then
$\varphi$ must be psh. This offers a version of converse to Berndtsson’s theorem
concerning log-plurisubharmonic variation of Bergman kernels and generalizes
a previous result of Li-Zhou. Additionally, we obtain a parallel result for the
positivity of Hermitian holomorphic line bundles on Kähler manifolds.
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...
We prove that, for every $n\ge1$, the degree-$n$ Bernstein operator on $[0,\pi]$ preserves positive-definiteness on the circle $S^1$. Equivalently, if a continuous function on $[0,\pi]$ defines a positive-definite isotropic kernel on $S^1$, then its Bernstein polynomial approximation of any fixed degree does as well. T...
Matthew Otten, Nam Nguyen, Thomas W. Watts· 0 citations
Let $(M,g,J)$ be a closed K\"ahler manifold satisfying $\operatorname{Ric}\geqslant g$. We establish improved Liouville theorems for the Euler--Lagrange equations associated with the Beckner--Sobolev inequalities by incorporating the first positive eigenvalue of the $\bar\partial$-Laplacian into a differential-identity...
We prove that if the $n$-dimensional torus $T^n$ is smoothly immersed in the unit ball $B^q\subset\mathbb R^q$ and $n\leq 18$, then there exists a point at which the spherical average of $\lvert II(u,u)\rvert^2$ is at least $3n/(n+2)$. This answers a question of Petrunin in these dimensions. The proof combines the scal...
Let $M$ be a $\mathrm{II}_1$ factor, $N$ a tracial von Neumann algebra, and $\Phi: M \rightarrow N$ a subtracial completely positive map. For an irreducible $\mathrm{II}_1$ subfactor $P \subseteq M$, we characterize when $\Phi$ exhibits a flattening property under conjugation by unitaries in $P$. To be specific, we sho...
We prove that if a real-valued function $f\in L^1$ on the complex unit circle has a gap of width at least $\pi$ in its essential range, then $\exp(\tilde f)$ is not integrable, where $\tilde f$ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function $Y$ sat...
D. Norrbo, J. Virtanen· 0 citations
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