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 volume of standard $n$-sphere. This confirms a conjecture by Bray in 1997.
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. The proof reduces the problem to the nonnegativity of the cosine coefficients of the Bernstein images $Q_{n,m}=B_n[\cos(mx)]$, which we prove using an explicit coefficient formula and a two-regime positivity argument. We also discuss the higher-dimensional sphere analogue and show that the naive affine Bernstein operator fails to preserve the positive-definite cone already on $S^2$.
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 argument. As a consequence, we obtain improved Sobolev and Beckner inequalities that refine the known Riemannian and K\"ahler estimates when the first eigenvalue is sufficiently large. We also derive new upper bounds for the diameter of $(M,g)$.
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 scalar-curvature obstruction for the torus with a conformal-Laplacian argument.
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 show that the failure of a Pimsner-Popa type inequality for $E_P \circ \Phi^* \circ \Phi$ is the precise obstruction, equivalently characterized by left weak mixing of a naturally associated $P$-$N$ bimodule. As an application, we obtain an asymptotic orthogonalization result generalizing a result of Popa.
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 $\exp(\widetilde f)$ is not integrable, where $\widetilde f$ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function $Y$ satisfying $$ \int_0^\infty Y(x) e^{-x}\, dx = \infty. $$ We also establish a local version on arcs of the unit circle: Under a natural condition on the inverse images of the two sides of the gap, $\widetilde f$ fails to be $Y$-integrable on the arc; in particular, it suffices that one of these inverse images is not an interval modulo null sets. Finally, we obtain corresponding results for complex-valued functions.
Unknown authors· 0 citations
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