Monatshefte fur Mathematik

Abstract V-static metrics generalise the notion of static metrics, and stem from the work of Miao and Tam (Calc Var Partial Differ Equ 36(2):141–171, 2009), and Corvino, Eichmair, and Miao (Math Ann 357(2):551–584, 2013) on critical points of the volume functional over the space of compact manifolds with constant scalar curvature. In this article we show that these V-static metrics arise naturally in the context of asymptotically hyperbolic manifolds as critical points of the volume-renormalised mass, recently introduced by Dahl, Kröncke, and McCormick (A volume-renormalized mass for asymptotically hyperbolic manifolds. arXiv preprint arXiv:2307.06196, 2023). In particular, we show that critical points of the volume-renormalised mass over the space of constant scalar curvature asymptotically hyperbolic manifolds without boundary, or satisfying appropriate boundary conditions, are exactly V-static metrics. This is directly analogous to the relationship between critical points of the ADM mass and static metrics for asymptotically flat manifolds.

Abstract In December 2022, Da Lio, Gianocca, and Rivière developed a new theory to prove the upper semi-continuity of the sum of the Morse index and the nullity in geometric analysis (Da Lio, Gianocca, and Rivière in Morse index stability for critical points to conformally invariant Lagrangians, 2023, arXiv:2212.03124), and applied it to conformally invariant problems in dimension 2—which include harmonic maps. Their method, quickly extended to Willmore immersions and later to other geometric settings, is applied in this article to the Morse stability of biharmonic maps in critical dimension 4. A key step in the proof is to obtain an energy quantization in $$L^{2,1}$$ L 2 , 1 (the pre-dual of the Marcinkiewicz space $$L^{2,\infty }$$ L 2 , ∞ of weakly squared-integrable functions), and we prove a general result to establish this strong energy quantization, which allows us to recover previous results in a unified fashion.

Abstract Generalizing a result of Berndtsson and Charpentier, we provide sufficient conditions for $$L^2$$ L 2 Sobolev regularity of the Bergman projection acting on $$L^2$$ L 2 sections of a holomorphic line bundle restricted to a relatively compact domain with Lipschitz boundary in a Hermitian manifold. We provide examples to show that our methods work for domains in Hopf manifolds endowed with a suitable Hermitian metric.

Abstract In this paper we construct the fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation approach. In addition, we characterize the powers of the fractional sub-Laplacian in the Heisenberg group, and as a byproduct we compute the k-th order momenta with respect to the heat kernel.

The Pál inequality is a classical result which asserts that among all planar convex sets of given width the equilateral triangle is the one of minimal area. In this paper we prove three quantitative versions of this inequality, by quantifying how the closeness of the area of a convex set, of certain width, to the minimal value implies its closeness to the equilateral triangle. As a by-product, we also present a novel result concerning a quantitative inequality for the inradius of a set, under minimal width constraint.

In this note, we establish an asymptotic formula for the Segre classes of higher symmetric powers of vector bundles. As an application, we show that a sufficiently higher symmetric power of the tangent bundle of a Campana–Peternell manifold is big.

Let $$(X,d,\mu )$$ be a space of homogeneous type in the sense of Coifman–Weiss, which satisfies a n-doubling property with $$n>1$$ , and supports an $$L^2$$ -Poincaré inequality. Consider the time independent Schrödinger operator $$\mathscr {L}=\mathcal {L}+V$$ on X, where $$\mathcal {L}$$ is a non-negative operator generalized by a Dirichlet form, and V is a non-negative Muckenhoupt weight which admits a reverse Hölder inequality of order q for some $$q>\max \{1,n/2\}$$ . Without the $$C^1$$ -regularity hypothesis, we derive that a solution u to the parabolic Schrödinger equation $$\partial _tu+\mathscr {L}u=0$$ on $$X\times \mathbb {R}_+$$ satisfies the Carleson measure condition $$\begin{aligned} \sup _{B(x_B,r_B)}\frac{1}{\mu (B(x_B,r_B))}\int _{0}^{r^2_B}\int _{B(x_B,r_B)}(|t\partial _tu|^2+|\sqrt{t}\nabla _x u|^2)\textrm{d}\mu \frac{\textrm{d}t}{t}<\infty \end{aligned}$$ if and only if u can be represented as the Gaussian integral of a $$\textrm{BMO}_\mathscr {L}$$ -function f. As an application, some limiting behaviors of the Carleson measure/BMO function are also considered.