The squared distance curvature is a kind of two-point curvature the sign of which turned out crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, an indirect one via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.

We show that quasimap Floer cohomology for varying symplectic quotients resolves several puzzles regarding displaceability of toric moment fibers. For example, we (i) present a compact Hamiltonian torus action containing an open subset of non-displaceable orbits and a codimension four singular set, partly answering a question of McDuff, and (ii) determine displaceability for most of the moment fibers of a symplectic ellipsoid.

In this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randers metric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

We study the geometry of non-reductive $4$-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable.

We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$, while, in the case where $M$ is of odd dimension, our objective is achieved by using a convenient almost cosymplectic structure. Several examples and applications are presented.

We consider the Randers space $(V^n,F_b)$ obtained by perturbing the Euclidean metric by a translation, $F_b=\alpha+\beta$, where $\alpha$ is the Euclidean metric and $\beta$ is a $1$-form with norm $b$, $0\leq b\lt 1$. We introduce the concept of a hypersurface with constant mean curvature in the direction of a unitary normal vector field. We obtain the ordinary differential equation that characterizes the rotational surfaces $(V^3,F_b)$ of constant mean curvature (cmc) in the direction of a unitary normal vector field. These equations reduce to the classical equation of the rotational cmc surfaces in Euclidean space, when $b=0$. It also reduces to the equation that characterizes the minimal rotational surfaces in $(V^3,F_b)$ when $H=0$, obtained by M. Souza and K. Tenenblat. Although the differential equation depends on the choice of the normal direction, we show that both equations determine the same rotational surface, up to a reflection. We also show that the round cylinders are cmc surfaces in the direction of the unitary normal field. They are generated by the constant solution of the differential equation. By considering the equation as a nonlinear dynamical system, we provide a qualitative analysis, for $0\lt b\lt \frac{\sqrt{3}}{3}$. Using the concept of stability and considering the linearization around the single equilibrium point (the constant solution), we verify that the solutions are locally asymptotically stable spirals. This is proved by constructing a Lyapunov function for the dynamical system and by determining the basin of stability of the equilibrium point. The surfaces of rotation generated by such solutions tend asymptotically to one end of the cylinder.

Speyer and Sturmfels associated Gröbner toric degenerations $\mathrm{Gr}_2(\mathbb{C}^n)^{\mathcal{T}}$ of $\mathrm{Gr}_2(\mathbb{C}^n)$ with each trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations induce toric degenerations $M_{\mathbf{r}}^{\mathcal{T}}$ of $M_{\mathbf{r}}$, the space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers and describe the action of the compact part of the torus as "bendings of polygons". We prove the conjecture of Foth and Hu that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida.

We construct a generalization of Courant algebroids that are classified by the third cohomology group $H^3(A,V)$, where $A$ is a Lie Algebroid, and $V$ is an $A$-module. We see that both Courant algebroids and $\mathcal{E}^1(M)$ structures are examples of them. Finally we introduce generalized CR structures on a manifold, which are a generalization of generalized complex structures, and show that every CR structure and contact structure is an example of a generalized CR structure.

Perelman established a differential Li--Yau--Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds. As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete noncompact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. The conditions are satisfied by asymptotically flat manifolds. We also prove a long time existence result for the K\"ahler--Ricci flow on complete nonnegatively curved K\"ahler manifolds.

Let $L$ be an oriented Lagrangian submanifold in an $n$-dimensional Kähler manifold~$M$. Let $u \colon D \to M$ be a minimal immersion from a disk $D$ with $u(\partial D) \subset L$ such that $u(D)$ meets $L$ orthogonally along $u(\partial D)$. Then the real dimension of the space of admissible holomorphic variations is at least $n+\mu(E,F)$, where $\mu(E,F)$ is a boundary Maslov index; the minimal disk is holomorphic if there exist $n$ admissible holomorphic variations that are linearly independent over $\mathbb{R}$ at some point $p \in \partial D$; if $M = \mathbb{C}P^n$ and $u$ intersects $L$ positively, then $u$ is holomorphic if it is stable, and its Morse index is at least $n+\mu(E,F)$ if $u$ is unstable.

A theorem of Tietze and Nakajima, from 1928, asserts that if a subset $X$ of $\mathbb{R}^n$ is closed, connected, and locally convex, then it is convex. We give an analogous ``local to global convexity" theorem when the inclusion map of $X$ to $\mathbb{R}^n$ is replaced by a map from a topological space $X$ to $\mathbb{R}^n$ that satisfies certain local properties. Our motivation comes from the Condevaux--Dazord--Molino proof of the Atiyah--Guillemin--Sternberg convexity theorem in symplectic geometry.

{Correspondence} between torsion-free connections with {nilpotent skew-symmetric curvature operator} and IP Riemann extensions is shown. Some consequences are derived in the study of four-dimensional IP metrics and locally homogeneous affine surfaces.

We address the problem of computing the fundamental group of a symplectic $S^1$-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian $S^1$-action. Several examples are presented to illustrate our main results.

The space of Monge-Ampère functions, introduced by J. H. G. Fu, is a space of rather rough functions in which the map $u\mapsto \operatorname{Det} D^2 u$ is well defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge-Ampère functions. We also prove that if a Monge-Ampère function $u$ on a bounded set $\Omega\subset\mathcal{R}^2$ satisfies the equation $\operatorname{Det} D^2 u=0$ in a particular weak sense, then the graph of $u$ is a developable surface, and moreover $u$ enjoys somewhat better regularity properties than an arbitrary Monge-Ampère function of $2$ variables.

Let $( X,g) $ be a complete noncompact Kähler manifold, of dimension $n\geq2,$ with positive Ricci curvature and of standard type (see the definition below). N. Mok proved that $X$ can be compactified, \emph{i.e.,} $X$ is biholomorphic to a quasi-projective variety$.$ The aim of this paper is to prove that the $L^{2}$ holomorphic sections of the line bundle $K_{X}^{-q}$ and the volume form of the metric $g$ have no essential singularities near the divisor at infinity. As a consequence we obtain a comparison between the volume forms of the Kähler metric $g$ and of the Fubini--Study metric induced on $X$. In the case of $\dim_{\mathbb{C} }X=2,$ we establish a relation between the number of components of the divisor $D$ and the dimension of the groups $H^{i}( \overline{X}, \Omega_{\overline{X}}^{1}( \log D) )$.

In this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann--Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions $2$ and $3$. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing S-curvature. This means that reversible non-Berwaldian Finsler spaces with vanishing S-curvature may exist at large. Hence the generalized volume comparison theorems due to Z. Shen are valid for a rather large class of Finsler spaces.

A Riemannian manifold $(M,\rho)$ is called Einstein if the metric $\rho$ satisfies the condition \linebreak$\Ric (\rho)=c\cdot \rho$ for some constant $c$. This paper is devoted to the investigation of $G$-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces $G/H$ of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds $\SO(n)/\SO(l)$. Furthermore, we show that for any positive integer $p$ there exists a Stiefel manifold $\SO(n)/\SO(l)$ that admits at least $p$ $\SO(n)$-invariant Einstein metrics.

In this paper, we study a long existing open problem on Landsberg metrics in Finsler geometry. We consider Finsler metrics defined by a Riemannian metric and a $1$-form on a manifold. We show that a \emph{regular} Finsler metric in this form is Landsbergian if and only if it is Berwaldian. We further show that there is a two-parameter family of functions, $\phi=\phi(s)$, for which there are a Riemannian metric $\alpha$ and a $1$-form $\beta$ on a manifold $M$ such that the scalar function $F=\alpha \phi (\beta/\alpha)$ on $TM$ is an almost regular Landsberg metric, but not a Berwald metric.

Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if $\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We prove the converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0) realization of a Coxeter group $W$, and $\Gamma$ is a subgroup of $W$. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings.

We discuss the subRiemannian geometry induced by two noncommutative vector fields which are left invariant on the Lie group $\mathbb{S}^3$.

For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form $\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we show that if the mean curvature vector of $M^n$ is parallel and the sectional curvature $K$ of $M^n$ satisfies some inequality, then the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is parallel and our manifold $M^n$ is a space form.

We consider a complete noncompact Riemannian manifold $M$ and give conditions on a compact submanifold $K \subset M$ so that the outward normal exponential map off the boundary of $K$ is a diffeomorphism onto $\MlK$. We use this to compactify $M$ and show that pinched negative sectional curvature outside $K$ implies $M$ has a compactification with a well-defined H\"older structure independent of $K$. The H\"older constant depends on the ratio of the curvature pinching. This extends and generalizes a 1985 result of Anderson and Schoen.

Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength $\epsilon$ of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$ (and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles $[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point value of the flow average of the leading perturbation.

In this paper, we find equations that characterize locally projectively flat Finsler metrics in the form $F = (\alpha + \beta)^2/\alpha$, where $\alpha=\sqrt{a_{ij}y^iy^j}$ is a Riemannian metric and $\beta= b_i y^i$ is a $1$-form. Then we completely determine the local structure of those with constant flag curvature.