We derive estimates relating the values of a solution at any two points to the distance between the points for quasilinear parabolic equations on compact Riemannian manifolds under Ricci flow.

We first introduce the weighted averaged projection sequence in $\text{CAT}(\unicode[STIX]{x1D705})$ spaces and then we establish some inequalities for the weighted averaged projection sequence. Using the inequalities, we prove the asymptotic regularity and the $\unicode[STIX]{x1D6E5}$-convergence of the weighted averaged projection sequence. Furthermore, we prove the strong convergence of the sequence under certain regularity or compactness conditions on $\text{CAT}(\unicode[STIX]{x1D705})$ spaces.

In this paper, we study Finsler warped product metrics with relatively isotropic Landsberg curvature. We obtain the differential equations that characterize such metrics. Then we give some examples.

A Willmore surface $y:M\rightarrow S^{n+2}$ has a natural harmonic oriented conformal Gauss map $Gr_{y}:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$ , which maps each point $p\in M$ to its oriented mean curvature 2-sphere at $p$ . An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a restricted nilpotency condition, which will be called “strongly conformally harmonic.” The goal of this paper is to characterize those strongly conformally harmonic maps from a Riemann surface $M$ to $SO^{+}(1,n+3)/SO^{+}(1,3)\times SO(n)$ , which are the conformal Gauss maps of some Willmore surface in $S^{n+2}.$ It turns out that generically, the condition of being strongly conformally harmonic suffices to be associated with a Willmore surface. The exceptional case will also be discussed.

We show that a class of area-preserving flows can deform every starshaped curve into a circle.

In this note, we prove that a four-dimensional compact oriented half-conformally flat Riemannian manifold M4 is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2}$, provided that the sectional curvatures all lie in the interval $\left[ {{{3\sqrt {3 - 5} } \over 4}, 1} \right]$ In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the 4-sphere.

We study immersed surfaces in ${\mathbb R}^3$ that are critical points of the Willmore functional under boundary constraints. The two cases considered are when the surface meets a plane orthogonally along the boundary and when the boundary is contained in a line. In both cases we derive weak forms of the resulting free boundary conditions and prove regularity by reflection.

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find conditions on the second fundamental form that characterise the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension two surfaces.

This work is concerned with Bielawski’s hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group $G$, a reductive subgroup $H\subseteq G$, and a Slodowy slice $S\subseteq \mathfrak{g}:=\text{Lie}(G)$, defining it to be the hyperkähler quotient of $T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of $G$. This hyperkähler slice is empty in some of the most elementary cases (e.g., when $S$ is regular and $(G,H)=(\text{SL}_{n+1},\text{GL}_{n})$, $n\geqslant 3$), prompting us to seek necessary and sufficient conditions for non-emptiness.

We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$-regularity of $(G,H)$. This $\mathfrak{a}$-regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$. We also provide a classification of the $\mathfrak{a}$-regular pairs $(G,H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces.

Let $M$ be a topological spherical space form, i.e., a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

Given a manifold $M$ with a submanifold $N$, the deformation space ${\mathcal{D}}(M,N)$ is a manifold with a submersion to $\mathbb{R}$ whose zero fiber is the normal bundle $\unicode[STIX]{x1D708}(M,N)$, and all other fibers are equal to $M$. This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with $N$ a submanifold transverse to the foliation. New examples include $L_{\infty }$-algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theorem around $N$, in terms of a model structure over $\unicode[STIX]{x1D708}(M,N)$.

In this paper we study topological properties of the right action by translation of the Weyl chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing.

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.

As was shown by a part of the authors, for a given $(2,3,5)$-distribution $D$ on a five-dimensional manifold $Y$, there is, locally, a Lagrangian cone structure $C$ on another five-dimensional manifold $X$ which consists of abnormal or singular paths of $(Y,D)$. We give a characterization of the class of Lagrangian cone structures corresponding to $(2,3,5)$-distributions. Thus, we complete the duality between $(2,3,5)$-distributions and Lagrangian cone structures via pseudo-product structures of type $G_{2}$. A local example of nonflat perturbations of the global model of flat Lagrangian cone structure which corresponds to $(2,3,5)$-distributions is given.

In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions. Starting with a Finsler spray $S$ and a holonomy invariant function ${\mathcal{P}}$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2\unicode[STIX]{x1D706}{\mathcal{P}}{\mathcal{C}}$. We prove that for any holonomy invariant nontrivial function ${\mathcal{P}}$ and for almost every value $\unicode[STIX]{x1D706}\in \mathbb{R}$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray. In these cases, the holonomy invariant function ${\mathcal{P}}$ is necessarily one of the principal curvatures of the geodesic structure.

Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an application, we prove the existence of global moduli spaces of scalar-flat Kähler ALE metrics for several infinite families of Kähler ALE spaces.

We prove the existence of a one-parameter family of nearly parallel G2-structures on the manifold $\text{S}^{3}\times \mathbb{R}^{4}$, which are mutually non-isomorphic and invariant under the cohomogeneityone action of the group SU(2)3. This family connects the two locally homogeneous nearly parallel G2-structures that are induced by the homogeneous ones on the sphere S7.

The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi-Riemannian and magnetic structures on manifolds. It is an extended report on problem sessions held during the International Workshop on Geodesics in August 2010 at the Chern Institute of Mathematics in Tianjin.

The simplicial complexity is an invariant for finitely presentable groups which was recently introduced by Babenko, Balacheff, and Bulteau to study systolic area. The simplicial complexity κ(G) was proved to be a good approximation of the systolic area σ(G) for large values of κ(G). In this paper we compute the simplicial complexity of all surface groups (both in the orientable and in the non-orientable case). This partially settles a problem raised by Babenko, Balacheff, and Bulteau. We also prove that κ(G * ℤ) = κ(G) for any surface group G. This provides the first partial evidence in favor of the conjecture of the stability of the simplicial complexity under free product with free groups. The general stability problem, both for simplicial complexity and for systolic area, remains open.

In this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold $G/P$. The key to this approach is that in each case $G/P$ is the homogeneous model for a parabolic geometry; the theory of such geometries provides a large supply of geometric tools and invariant differential operators that can be used for this study. A classical theorem of Wolf shows that any involutive automorphism of a semisimple Lie group $G$ with fixed point group $H$ gives rise to a large family of such compactifications of homogeneous spaces of $H$. Most examples of (classical) Riemannian symmetric spaces as well as many non-symmetric examples arise in this way. A specific feature of the approach is that any compactification of that type comes with the notion of ‘curved analog’ to which the tools we develop also apply. The model example of this is a general Poincaré–Einstein manifold forming the curved analog of the conformal compactification of hyperbolic space. In the first part of the article, we derive general tools for the analysis of such compactifications. In the second part, we analyze two families of examples in detail, which in particular contain compactifications of the symmetric spaces $\mathit{SL}(n,\mathbb{R})/\mathit{SO}(p,n-p)$ and $\mathit{SO}(n,\mathbb{C})/\mathit{SO}(n)$. We describe the decomposition of the compactification into orbits, show how orbit closures can be described as the zero sets of smooth solutions to certain invariant differential operators and prove a local slice theorem around each orbit in these examples.