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.

Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle ${\mathcal{O}}_{T_{M}^{\ast }}(1)$ is not pseudo-effective and ${\mathcal{O}}_{T_{N}^{\ast }}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

Paraconformal or GL(2, ℝ) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree n – 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source of GL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context, GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.

Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutive GL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-free GL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesic GL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesic α-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.

Our main result states that involutive GL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n – 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

The main result of this paper is the following: any weighted Riemannian manifold $(M,g,\unicode[STIX]{x1D707})$, i.e., a Riemannian manifold $(M,g)$ endowed with a generic non-negative Radon measure $\unicode[STIX]{x1D707}$, is infinitesimally Hilbertian, which means that its associated Sobolev space $W^{1,2}(M,g,\unicode[STIX]{x1D707})$ is a Hilbert space.

We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold $(M,F,\unicode[STIX]{x1D707})$ can be isometrically embedded into the space of all measurable sections of the tangent bundle of $M$ that are $2$-integrable with respect to $\unicode[STIX]{x1D707}$.

By following the same approach, we also prove that all weighted (sub-Riemannian) Carnot groups are infinitesimally Hilbertian.

We prove the analogue of the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball $\mathbb{B}^{n}$ using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ and the associated uniformization map $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$. Given an algebraic subset $S\,\subset \,\mathbb{B}^{n}$ and writing $Z$ for the Zariski closure of $\unicode[STIX]{x1D70B}(S)$ in $X_{\unicode[STIX]{x1D6E4}}$ (which is equipped with a canonical quasi-projective structure), in some precise sense we realize $Z$ as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component $\widetilde{Z}$ of $\unicode[STIX]{x1D70B}^{-1}(Z)$ as $\widetilde{Z}$ exits the boundary $\unicode[STIX]{x2202}\mathbb{B}^{n}$ by exploiting the strict pseudoconvexity of $\mathbb{B}^{n}$, culminating in the proof that $\widetilde{Z}\,\subset \,\mathbb{B}^{n}$ is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of $\text{ Aut}(\unicode[STIX]{x1D6FA})$ for (possibly reducible) bounded symmetric domains $\unicode[STIX]{x1D6FA}$.

We show that almost stable constant mean curvature hypersurfaces contained in a sufficiently small ball of a manifold of bounded sectional curvature are close to geodesic spheres.

Let $(M,I,J,K,g)$ be a hyperkähler manifold. Then the complex manifold $(M,I)$ is holomorphic symplectic. We prove that for all real $x,y$, with $x^{2}+y^{2}=1$ except countably many, any finite-energy $(xJ+yK)$-holomorphic curve with boundary in a collection of $I$-holomorphic Lagrangians must be constant. By an argument based on the Łojasiewicz inequality, this result holds no matter how the Lagrangians intersect each other. It follows that one can choose perturbations such that the holomorphic polygons of the associated Fukaya category lie in an arbitrarily small neighborhood of the Lagrangians. That is, the Fukaya category is local. We show that holomorphic Lagrangians are tautologically unobstructed. Moreover, the Fukaya $A_{\infty }$ algebra of a holomorphic Lagrangian is formal. Our result also explains why the special Lagrangian condition holds without instanton corrections for holomorphic Lagrangians.

We construct a cycle in higher Hochschild homology associated to the two-dimensional torus which represents 2-holonomy of a nonabelian gerbe in the same way as the ordinary holonomy of a principal G-bundle gives rise to a cycle in ordinary Hochschild homology. This is done using the connection 1-form of Baez–Schreiber. A crucial ingredient in our work is the possibility to arrange that in the structure crossed module $\unicode[STIX]{x1D707}:\mathfrak{h}\rightarrow \mathfrak{g}$ of the principal 2-bundle, the Lie algebra $\mathfrak{h}$ is abelian, up to equivalence of crossed modules.

We prove that among all compact homogeneous spaces for an effective transitive action of a Lie group whose Levi subgroup has no compact simple factors, the seven-dimensional flat torus is the only one that admits an invariant torsion-free $\text{G}_{2(2)}$-structure.

We answer Mark Kac’s famous question, “Can one hear the shape of a drum?” in the positive for orbifolds that are 3-dimensional and 4-dimensional lens spaces; we thus complete the answer to this question for orbifold lens spaces in all dimensions. We also show that the coefficients of the asymptotic expansion of the trace of the heat kernel are not sufficient to determine the above results.

In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) related to the radial derivation (i.e., the derivation along the geodesic curves) on the Cartan–Hadamard manifolds. By Gauss lemma, our new Hardy inequalities are stronger than the classical ones. We also establish the improvements of these inequalities in terms of sectional curvature of the underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequalities on the hyperbolic space ℍn. Especially, we show that our new Rellich inequalities are indeed stronger than the classical ones on the hyperbolic space ℍn.

We extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic simply connected and positively-curved souls. Examples of such pairs are given by explicit pairs of Eschenburg spaces. To deduce the second statement from the first, we extend our earlier work on the stable converse soul question and show that it has a positive answer for a class of spaces that includes all Eschenburg spaces.

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a group action. We also prove another topological rigidity result for hypersurfaces of the sphere that involves the spherical image of its usual Gauss map.

We introduce the notion of Lie invariant structure Jacobi operators for real hypersurfaces in the complex quadric $Q^{m}=SO_{m+2}/SO_{m}SO_{2}$. The existence of invariant structure Jacobi operators implies that the unit normal vector field $N$ becomes $\mathfrak{A}$-principal or $\mathfrak{A}$-isotropic. Then, according to each case, we give a complete classification of real hypersurfaces in $Q^{m}=SO_{m+2}/SO_{m}SO_{2}$ with Lie invariant structure Jacobi operators.

A new method for classifying naturally reductive spaces is presented. This method relies on a new construction and the structure theory of naturally reductive spaces recently developed by the author. This method is applied to obtain the classification of all naturally reductive spaces in dimension 7 and 8.