For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev–Zelenko–Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet–Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension $4d+3$, with $d>1$, has sub-Riemannian diameter bounded by $\unicode[STIX]{x1D70B}$. When $d=1$, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on $\mathbb{S}^{4d+3}$ of the quaternionic Hopf fibrations:

$$\begin{eqnarray}\mathbb{S}^{3}{\hookrightarrow}\mathbb{S}^{4d+3}\rightarrow \mathbb{HP}^{d},\end{eqnarray}$$

$\unicode[STIX]{x1D70B}$

$d\geqslant 1$

Extending our previous work on eigenvalues of closed surfaces and work of Otal and Rosas, we show that a complete Riemannian surface $S$ of finite type and Euler characteristic $\unicode[STIX]{x1D712}(S)<0$ has at most $-\unicode[STIX]{x1D712}(S)$ small eigenvalues.

Let $M$ be an irreducible holomorphic symplectic (hyperkähler) manifold. If $b_{2}(M)\geqslant 5$ , we construct a deformation $M^{\prime }$ of $M$ which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real $(1,1)$ -classes is hyperbolic. If $b_{2}(M)\geqslant 14$ , similarly, we construct a deformation which admits a parabolic automorphism (and many other automorphisms as well).

We apply a mean-value inequality for positive subsolutions of the $f$ -heat operator, obtained from a Sobolev embedding, to prove a nonexistence result concerning complete noncompact $f$ -maximal spacelike hypersurfaces in a class of weighted Lorentzian manifolds. Furthermore, we establish a new Calabi–Bernstein result for complete noncompact maximal spacelike hypersurfaces in a Lorentzian product space.

We give a new and simple proof of a result of Ding and Xin, which states that any smooth complete self-shrinker in $\mathbb{R}^{3}$ with the second fundamental form of constant length must be a generalised cylinder $\mathbb{S}^{k}\times \mathbb{R}^{2-k}$ for some $k\leq 2$ . Moreover, we prove a gap theorem for smooth self-shrinkers in all dimensions.

Let $X$ be a compact 4-manifold with boundary. We study the space of hyperkähler triples $\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3}$ on $X$ , modulo diffeomorphisms which are the identity on the boundary. We prove that this moduli space is a smooth infinite-dimensional manifold and describe the tangent space in terms of triples of closed anti-self-dual 2-forms. We also explore the corresponding boundary value problem: a hyperkähler triple restricts to a closed framing of the bundle of 2-forms on the boundary; we identify the infinitesimal deformations of this closed framing that can be filled in to hyperkähler deformations of the original triple. Finally we study explicit examples coming from gravitational instantons with isometric actions of $\text{SU}(2)$ .

Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C 2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.

This paper considers the Ricci flow coupled with the harmonic map flow between two manifolds. We derive estimates for the fundamental solution of the corresponding conjugate heat equation and we prove an analogue of Perelman's differential Harnack inequality. As an application, we find a connection between the entropy functional and the best constant in the Sobolev embedding theorem in ℝn .

A famous conjecture of Hopf states that $\mathbb{S}^{2}\times \mathbb{S}^{2}$ does not admit a Riemannian metric with positive sectional curvature. In this article, we prove that no manifold product $N\times N$ can carry a metric of positive sectional curvature admitting a certain degree of torus symmetry.

We develop new algorithms for approximating extremal toric Kähler metrics. We focus on an extremal metric on , which is conformal to an Einstein metric (the Chen–LeBrun–Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various geometric quantities. In particular, we demonstrate a small eigenvalue of the scalar Laplacian of the Einstein metric that gives numerical evidence that the Einstein metric is conformally unstable under Ricci flow.

We study the second-order invariants of a Lorentzian surface in ℝ2,2, and the curvature hyperbolas associated with its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentzian surface and give an extrinsic proof of the vanishing of the total Gauss and normal curvatures of a compact Lorentzian surface. The Gauss map and the second-order invariants are then used to study the asymptotic directions of a Lorentzian surface and discuss their causal character. We also consider the relation of the asymptotic lines with the mean directionally curved lines. We finally introduce and describe the quasi-umbilic surfaces, and the surfaces whose four classical invariants vanish identically.

We prove a DDVV inequality for submanifolds of warped products of the form $I\times _{a}\mathbb{M}^{n}(c)$ , where $I$ is an interval and $\mathbb{M}^{n}(c)$ is a real space form of curvature $c$ . As an application, we give a rigidity result for submanifolds of $\mathbb{R}\times _{e^{\unicode[STIX]{x1D706}t}}\mathbb{H}^{n}(c)$ .

We give a classification of Levi-umbilical real hypersurfaces in a complex space form $\widetilde{M}_{n}(c)$ , $n\geqslant 3$ , whose Levi form is proportional to the induced metric by a nonzero constant. In a complex projective plane $\mathbb{C}\mathbb{P}^{2}$ , we give a local construction of such hypersurfaces and moreover, we give new examples of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$ .

In this paper, we establish new characterization results concerning totally umbilical hypersurfaces of the hyperbolic space $\mathbb{H}^{n+1}$ , under suitable constraints on the behavior of the Lorentzian Gauss map of complete hypersurfaces having some constant higher order mean curvature. Furthermore, working with different warped product models for $\mathbb{H}^{n+1}$ and supposing that certain natural inequalities involving two consecutive higher order mean curvature functions are satisfied, we study the rigidity and the nonexistence of complete hypersurfaces immersed in $\mathbb{H}^{n+1}$ .

In this paper, we consider contact metric three-manifolds $(M;\unicode[STIX]{x1D702},g,\unicode[STIX]{x1D711},\unicode[STIX]{x1D709})$ which satisfy the condition $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}h=\unicode[STIX]{x1D707}h\unicode[STIX]{x1D711}+\unicode[STIX]{x1D708}h$ for some smooth functions $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D708}$ , where $2h=\unicode[STIX]{x00A3}_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D711}$ . We prove that if $M$ is conformally flat and if $\unicode[STIX]{x1D707}$ is constant, then $M$ is either a flat manifold or a Sasakian manifold of constant curvature $+1$ . We cannot extend this result for a smooth function $\unicode[STIX]{x1D707}$ . Indeed, we give such an example of a conformally flat contact three-manifold which is not of constant curvature.

The Lusternik–Schnirelmann category of the quaternionic Grassmannian

is known to be k(n − k). In this paper we show that this result can be deduced from Morse theory.

The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on $\mathbb{C}\text{P}^{2}$ with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a $\text{CAT}[0]$ ramification and prove this in several cases. In the latter case we prove that the ramification is $\text{CAT}[0]$ if the metric on $\mathbb{C}\text{P}^{2}$ is non-negatively curved. We deduce that complex line arrangements in $\mathbb{C}\text{P}^{2}$ studied by Hirzebruch have aspherical complement.

We prove the existence of weak solutions of complex $m$ -Hessian equations on compact Hermitian manifolds for the non-negative right-hand side belonging to $L^{p}$ , $p>n/m$ ( $n$  is the dimension of the manifold). For smooth, positive data the equation has recently been solved by Székelyhidi and Zhang. We also give a stability result for such solutions.

We apply the lifting theorem of Searle and the second author to put metrics of almost nonnegative curvature on the fake $\mathbb{R}P^{6}$ s of Hirsch and Milnor and on the analogous fake $\mathbb{R}P^{14}$ s.

We obtain all the solutions of types u(x, y) = f(x) + g(y) and u(x, y) = f(x)g(y) for three known mean-curvature-prescribed equations, namely, the capillary equation, the translating soliton equation and the two-dimensional analogue of the catenary.