Manuscripta Mathematica最新文献

In this note, we derive a Yau type gradient estimate for positive exponentially harmonic functions on Riemannian manifolds with compact boundary. As its application, we obtain a Liouville type theorem.

This paper focuses on Gromov hyperbolic characterizations of unbounded uniform domains. Let (Gsubsetneq mathbb {R}^n) be an unbounded domain. We prove that the following conditions are quantitatively equivalent: (1) G is uniform; (2) G is Gromov hyperbolic with respect to the quasihyperbolic metric and linearly locally connected; (3) G is Gromov hyperbolic with respect to the quasihyperbolic metric and there exists a naturally quasisymmetric correspondence between its Euclidean boundary and the punctured Gromov boundary equipped with a Hamenstädt metric (defined by using a Busemann function). As an application, we investigate the boundary quasisymmetric extensions of quasiconformal mappings, and of more generally rough quasi-isometries between unbounded domains with respect to the quasihyperbolic metrics.

We study a one-dimensional Lagrangian problem including the variational reformulation, derived in a recent work of Ambrosio–Baradat–Brenier, of the discrete Monge–Ampère gravitational model, which describes the motion of interacting particles whose dynamics is ruled by the optimal transport problem. The more general action-type functional we consider contains a discontinuous potential term related to the descending slope of the opposite squared distance function from a generic discrete set in (mathbb {R}^{d}). We exploit the underlying geometrical structure provided by the associated Voronoi decomposition of the space to obtain (C^{1,1})-regularity for local minimizers out of a finite number of shock times.

Let ({mathfrak g}) be a reductive Lie algebra and (mathfrak tsubset mathfrak g) a Cartan subalgebra. The (mathfrak t)-stable decomposition ({mathfrak g}=mathfrak toplus {mathfrak m}) yields a bi-grading of the symmetric algebra ({mathcal {S}}({mathfrak g})). The subalgebra ({mathcal {Z}}_{({mathfrak g},mathfrak t)}) generated by the bi-homogenous components of the symmetric invariants (Fin {mathcal {S}}({mathfrak g})^{mathfrak g}) is known to be Poisson commutative. Furthermore the algebra ({tilde{{mathcal {Z}}}}=textsf{alg}langle {mathcal {Z}}_{({mathfrak g},{mathfrak t})},{mathfrak t}rangle ) is also Poisson commutative. We investigate relations between ({tilde{{mathcal {Z}}}}) and Mishchenko–Fomenko subalgebras. In type A, we construct a quantisation of ({tilde{{mathcal {Z}}}}) making use of quantum Mishchenko–Fomenko algebras.

We give an alternative computation of the Betti and Hodge numbers for manifolds of OG6 type using the method of Ngô Strings introduced by de Cataldo, Rapagnetta, and Saccà.

Let (ell ) be a prime. If (textbf{G}) is a compact connected Lie group, or a connected reductive algebraic group in characteristic different from (ell ), and (ell ) is a good prime for (textbf{G}), we show that the number of weights of the (ell )-fusion system of (textbf{G}) is equal to the number of irreducible characters of its Weyl group. The proof relies on the classification of (ell )-stubborn subgroups in compact Lie groups.

In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of compact manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a compact and simply-connected n-dimensional manifold of positive Ricci curvature (text {Ric}ge n-1) has length (le n pi .) This improves the bound (8pi (n-1)) given by Rotman (Positive Ricci curvature and the length of a shortest periodic geodesic. arXiv:2203.09492, 2022).

The aim of this paper is to give an upper bound for the intrinsic diameter of a surface with boundary immersed in a conformally flat three dimensional Riemannian manifold in terms of the integral of the mean curvature and of the length of its boundary. Of particular interest is the application of the inequality to minimal surfaces in the three-sphere and in the hyperbolic space. Here the result implies an a priori estimate for connected solutions of Plateau’s problem, as well as a necessary condition on the boundary data for the existence of such solutions. The proof follows a construction of Miura and uses a diameter bound for closed surfaces obtained by Topping and Wu–Zheng.

In this paper, we consider the logarithmic elliptic equations with critical exponent

Here, the parameters (Nge 6), (lambda in {{mathbb {R}}}), (theta >0) and ( 2^*=frac{2N}{N-2} ) is the Sobolev critical exponent. We prove the existence of a sign-changing solution with exactly two nodal domain for an arbitrary smooth bounded domain (Omega subset {mathbb {R}}^{N}). When (Omega =B_R(0)) is a ball, we also construct infinitely many radial sign-changing solutions with alternating signs and prescribed nodal characteristic.

Let (Omega ) be a domain on the unit n-sphere ( {mathbb {S}}^n) and ( overset{{,}_circ }{g}) the standard metric of ({mathbb {S}}^n), (nge 3). We show that there exists a conformal metric g with vanishing scalar curvature (R(g)=0) such that ((Omega , g)) is complete if and only if the Bessel capacity ({mathcal {C}}_{alpha , q}({mathbb {S}}^nsetminus Omega )=0), where (alpha =1+frac{2}{n}) and (q=frac{n}{2}). Our analysis utilizes some well known properties of capacity and Wolff potentials, as well as a version of the Hopf–Rinow theorem for the divergent curves.