Totally real flat minimal surfaces in the hyperquadric

In this paper, we study geometry of totally real minimal surfaces in the complex hyperquadric $Q_{N-2}$, and obtain some characterizations of the harmonic sequence generated by these minimal immersions. For totally real flat surfaces that are minimal immersed in both $Q_{N-2}$ and $\mathbb{C}{P}^{N-1}$, we determine them for $N=4,5,6$, and give a classification theorem when they are Clifford solutions.     

Geometry of the Fisher–Rao metric on the space of smooth densities on a compact manifold

It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form for some smooth functions of the total volume . Here we determine the geodesics and the curvature of this metric and study geodesic and metric completeness.     

Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form , where and F is a positive, strictly monotone and 1‐homogeneous curvature function. In particular this class includes the mean curvature . We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews–McCoy–Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface.     

Rigidity for critical metrics of the volume functional

Geodesic balls in a simply connected space forms , or are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao–Tam critical metrics with connected boundary provided that the boundary of the manifold has a lower bound for the Ricci curvature. In the same spirit we also extend a rigidity theorem due to Boucher et al. 7 and Shen 18 to n‐dimensional static metrics with positive constant scalar curvature, which gives us a partial answer to the Cosmic no‐hair conjecture.     

Minimal kernels of Dirac operators along maps

Let M be a closed spin manifold and let N be a closed manifold. For maps and Riemannian metrics g on M and h on N, we consider the Dirac operator of the twisted Dirac bundle . To this Dirac operator one can associate an index in . If M is 2‐dimensional, one gets a lower bound for the dimension of the kernel of out of this index. We investigate the question whether this lower bound is obtained for generic tupels .     

On the structure of WDC sets

WDC sets in were recently defined as sublevel sets of DC functions (differences of convex functions) at weakly regular values. They form a natural and substantial generalization of sets with positive reach and still admit the definition of curvature measures. Using results on singularities of convex functions, we obtain regularity results on the boundaries of WDC sets. In particular, the boundary of a compact WDC set can be covered by finitely many DC surfaces. More generally, we prove that any compact WDC set M of topological dimension can be decomposed into the union of two sets, one of them being a k‐dimensional DC manifold open in M, and the other can be covered by finitely many DC surfaces of dimension . We also characterize locally WDC sets among closed Lipschitz domains and among lower‐dimensional Lipschitz manifolds. Finally, we find a full characterization of locally WDC sets in the plane.     

The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair consisting of a left‐invariant Riemannian metric g and a positive constant c such that , where is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that is solvable for some left‐invariant Riemannian metric g.     

Classification of surfaces in a pseudo‐sphere with 2‐type pseudo‐spherical Gauss map

In this article, we study submanifolds in a pseudo‐sphere with 2‐type pseudo‐spherical Gauss map. We give a characterization theorem for Lorentzian surfaces in the pseudo‐sphere with zero mean curvature vector in and 2‐type pseudo‐spherical Gauss map. We also prove that non‐totally umbilical proper pseudo‐Riemannian hypersurfaces in a pseudo‐sphere with non‐zero constant mean curvature has 2‐type pseudo‐spherical Gauss map if and only if it has constant scalar curvature. Then, for we obtain the classification of surfaces in with 2‐type pseudo‐spherical Gauss map. Finally, we give an example of surface with null 2‐type pseudo‐spherical Gauss map which does not appear in Riemannian case, and we give a characterization theorem for Lorentzian surfaces in with null 2‐type pseudo‐spherical Gauss map.     

On para‐Kähler Lie algebroids and contravariant pseudo‐Hessian structures

In this paper, we generalize all the results obtained on para‐Kähler Lie algebras in [3] to para‐Kähler Lie algebroids. In particular, we study exact para‐Kähler Lie algebroids as a generalization of exact para‐Kähler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra , the orbits of the action Φ of on given by are pseudo‐Hessian manifolds, where . We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.     

Keller–Osserman estimates and removability result for the anisotropic porous medium equation with gradient absorption term

We study “large” nonnegative solutions for a class of quasilinear equations model of which is We give a sufficient condition on the exponents and for the removability of isolated singularities.     

Multiple solutions for a class of quasilinear Schrödinger equations

In this paper, we study the following quasilinear Schrödinger equations of the form where , , . Some existence results for positive solutions, negative solutions and sequence of high energy solutions are obtained via a perturbation method.     

Expected intrinsic volumes and facet numbers of random beta‐polytopes

Let be i.i.d. random points in the d‐dimensional Euclidean space sampled according to one of the following probability densities: and We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of . Asymptotic formulae were obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when , respectively , we can also cover the models in which are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of and . One of the main tools used in the proofs is the Blaschke–Petkantschin formula.     

Conjugacy classes,characters and products of elements

Recently, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if for every of coprime order. Motivated by this result, we study the groups with the property that and those with the property that for every and every nontrivial of pairwise coprime order. We also consider several ways of weakening the hypothesis on x and y. While the result of Baumslag and Wiegold is completely elementary, some of our arguments here depend on (parts of) the classification of finite simple groups.     

Singular rational curves on elliptic K3 surfaces

We show that on every elliptic K3 surface there are rational curves ${(R_i)}_{i\in \mathbb{N}}$ such that $R_i^2\to \infty$, that is, of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to $\mathbb{P}({\mathrm{\Omega }}_X)$ is dense in the Zariski topology. As an application, we give a simple proof of a theorem of Kobayashi in the elliptic case, that is, there are no globally defined symmetric differential forms.     

Weighted estimates for bilinear fractional integral operators

Moen (2016) proved weighted estimates for the bilinear fractional integrals where . We improve his results when and consider the case . As a corollary we obtain a bilinear Stein–Weiss inequality where .     

Starshapedness of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings

We study solutions of the problem (0.1) where are open sets such that , , and f is a nonlinearity. Under different assumptions on f we prove that, if D₀ and D₁ are starshaped with respect to the same point , then the same occurs for every superlevel set of u.     

Real hypersurfaces in the complex quadric whose structure Jacobi operator is of Codazzi type

First we introduce the notion of structure Jacobi operator of Codazzi type for real hypersurfaces in the complex quadric . Next we give a complete classification of real hypersurfaces in with structure Jacobi operator of Codazzi type.     

Singular Hamiltonian elliptic systems with critical exponential growth in dimension two

We will focus on the existence of nontrivial solutions to the following Hamiltonian elliptic system where are numbers belonging to the interval [0, 2), V is a continuous potential bounded below on by a positive constant and the functions f and g possess exponential growth range established by Trudinger–Moser inequalities in Lorentz–Sobolev spaces. The proof involves linking theorem and a finite‐dimensional approximation.     

Lindelöf theorems for monotone Sobolev functions in Orlicz spaces on uniform domains

In this paper, we are concerned with Lindelöf type theorems for monotone (in the sense of Lebesgue) Sobolev functions u on a uniform domain satisfying where ? denotes the gradient, denotes the distance from z to the boundary , φ is of log‐type and ω is a weight function satisfying the doubling condition.