The Journal of Geometric Analysis最新文献

We show that every closed nonpositively curved surface satisfies Loewner’s systolic inequality. The proof relies on a combination of the Gauss–Bonnet formula with an averaging argument using the invariance of the Liouville measure under the geodesic flow. This enables us to find a disk with large total curvature around its center yielding a large area.

On a certain class of 16-dimensional manifolds a new class of Riemannian metrics, called octonionic Kähler, is introduced and studied. It is an octonionic analogue of Kähler metrics on complex manifolds and of HKT-metrics of hypercomplex manifolds. Then for this class of metrics an octonionic version of the Monge–Ampère equation is introduced and solved under appropriate assumptions. The latter result is an octonionic version of the Calabi–Yau theorem from Kähler geometry.

In this paper, we study the necessary and sufficient conditions for the existence of solutions to the discrete (L_p) Minkowski problem of log-concave functions in ({mathbb {R}}) when (pge 1).

We investigate the property of boundary rigidity for the projective structures associated to torsion-free affine connections on connected smooth manifolds with boundary. We show that these structures are generically boundary rigid, meaning that any automorphism of a generic projective structure that restricts to the identity on the boundary must itself be the identity. However, and in contrast with what happens for example for conformal structures, we show that there exist projective structures which are not boundary rigid. We characterise these non-rigid structures by the vanishing of a certain local projective invariant of the boundary.

In this paper, we prove restriction theorems for the Fourier–Laguerre transform and establish Strichartz estimates for the Schrödinger propagator (e^{-itL_alpha }) for the Laguerre operator (L_alpha =-Delta -sum _{j=1}^{n}(dfrac{2alpha _j+1}{x_j}dfrac{partial }{partial x_j})+dfrac{|x|^2}{4}), (alpha =(alpha _1,alpha _2,ldots ,alpha _n)in {(-frac{1}{2},infty )^n}) on (mathbb {R}_+^n) involving systems of orthonormal functions. The proof is based on a combination of some known dispersive estimate and the argument in Nakamura [Trans Am Math Soc 373(2), 1455–1476 (2020)] on torus. As an application, we obtain the global well-posedness for the nonlinear Laguerre–Hartree equation in Schatten space.

In this paper, we show that the (L^2)-optimal condition implies the (L^2)-divisibility of (L^2)-integrable holomorphic functions. As an application, we offer a new characterization of bounded (L^2)-domains of holomorphy with null thin complements using the (L^2)-optimal condition, which appears to be advantageous in addressing a problem proposed by Deng-Ning-Wang. Through this characterization, we show that a domain in a Stein manifold with a null thin complement, admitting an exhaustion of complete Kähler domains, remains Stein. By the way, we construct an (L^2)-optimal domain that does not admit any complete Kähler metric.

We study a conormal boundary value problem for a class of quasilinear elliptic equations in bounded domain (Omega ) whose coefficients can be degenerate or singular of the type (text {dist}(x, partial Omega )^alpha ), where (partial Omega ) is the boundary of (Omega ) and (alpha in (-1, infty )) is a given number. We establish weighted Sobolev type estimates for weak solutions under a smallness assumption on the weighted mean oscillations of the coefficients in small balls. Our approach relies on a perturbative method and several new Lipschitz estimates for weak solutions to a class of singular-degenerate quasilinear equations.

We find the local form of all non-closed Lorentzian Weyl manifolds ((M,c,nabla )) with recurrent curvature tensor. The recurrent curvature tensor turns out to be weighted parallel, i.e., the obtained spaces provide certain generalization of locally symmetric affine spaces for the Weyl geometry. If the dimension of the manifold is greater than 3, then the conformal structure is flat, and the recurrent Weyl structure is locally determined by a single function of one variable. Two local structures are equivalent if and only if the corresponding functions are related by a transformation from (textrm{Aff}^0_1(mathbb {R})times textrm{PSL}_2(mathbb {R})times {mathbb {Z}}_2). We find generators for the field of rational scalar differential invariants of this Lie group action. The global structure of the manifold M may be described in terms of a foliation with a transversal projective structure. It is shown that all locally homogeneous structures are locally equivalent, and there is only one simply connected homogeneous non-closed recurrent Lorentzian Weyl manifold. Moreover, there are 5 classes of cohomogeneity-one spaces, and all other spaces are of cohomogeneity-two. If (dim M=3), the non-closed recurrent Lorentzian Weyl structures are locally determined by one function of two variables or two functions of one variable, depending on whether its holonomy algebra is 1- or 2-dimensional. In this case, two structures with the same holonomy algebra are locally equivalent if and only if they are related, respectively, by a transformation from an infinite-dimensional Lie pseudogroup or a 4-dimensional subgroup of (textrm{Aff}({mathbb {R}}^3)). Again we provide generators for the field of rational differential invariants. We find a local expression for the locally homogeneous non-closed recurrent Lorentzian Weyl manifolds of dimension 3, and also of those of cohomogeneity one and two. In the end we give a local description of the non-closed recurrent Lorentzian Weyl manifolds that are also Einstein–Weyl. All of them are 3-dimensional and have a 2-dimensional holonomy algebra.

This article deals with the following fractional (frac{N}{s})-Laplace Kichhoff equation involving exponential growth of the form:

where (varepsilon >0) is a parameter, (sin (0,1)) and ((-Delta )_p^s) is the fractional p-Laplace operator with (p=frac{N}{s}ge 2), K is a Kirchhoff function, f is a continuous function with exponential growth and Z is a potential function possessing a local minimum. Under some suitable conditions, we obtain the existence, multiplicity and concentration of solutions to the above problem via penalization methods and Lyusternik-Schnirelmann theory.

In this paper, we look for normalized solutions to the following non-autonomous Schrödinger equation

where (Nge 3), (a>0), (lambda in {mathbb {R}} ), (hne const) and (2^*=frac{2N}{N-2}) is the Sobolev critical exponent. In the (L^2)-subcritical regime (i.e. (2<q<2+frac{4}{N})), by proposing some new conditions on h, we verify that the corresponding Pohozaev manifold is a natural constraint and establish the existence of normalized ground states. Compared to the (L^2)-subcritical regime, it is necessary to apply some reverse conditions to h provided that at least (L^2)-critical regime (i.e. (2+frac{4}{N}le q<2^*)) is considered. We prove the existence of minimizer on the Pohozaev manifold of the associated energy functional and determine that the minimizer is a normalized solution by using the classical deformation lemma. In particular, by imposing further assumptions on h, the ground states can be obtained.