The Chen–Teo gravitational instanton is an asymptotically flat, toric, Ricci-flat family of metrics on $\mathbb{CP}^2 \setminus S^1$, that provides a counter-example to the classical Euclidean Black Hole Uniqueness conjecture. In this paper we show that the Chen–Teo instanton is Hermitian and non-Kähler. It follows that all known examples of gravitational instantons are Hermitian.

Given a general Anosov $\mathbb{R}^\kappa$ action on a closed manifold, we study properties of certain invariant measures that have recently been introduced in $\href{https://doi.org/10.48550/arXiv.2007.14275}{[GBGHW20]}$ using the theory of Ruelle–Taylor resonances. We show that these measures share many properties of Sinai–Ruelle–Bowen measures for general Anosov flows such as smooth disintegrations along the unstable foliation, positive Lebesgue measure basins of attraction and a Bowen formula in terms of periodic orbits. Finally we show that if the action in the positive Weyl chamber is transitive, the measure is unique and has full support.

In this article, we show that there exist discrete isometry groups of the 2‑ and 3‑dimensional complex hyperbolic spaces with critical exponents arbitrarily close to but strictly smaller than the maximum possible value. This result shows no gap in the values of critical exponents for complex hyperbolic Kleinian groups.

Let $(M,g)$ be an asymptotically flat Riemannian manifold of dimension $n \geq 3$ with positive mass. We give a short proof based on Lyapunov–Schmidt reduction of the existence of an asymptotic foliation of $(M,g)$ by stable constant mean curvature spheres. Moreover, we show that the geometric center of mass of the foliation agrees with the Hamiltonian center of mass of $(M,g)$. In dimension $n = 3$, these results were shown previously by C. Nerz using a different approach. In the case where $n = 3$ and the scalar curvature of $(M,g)$ is nonnegative, we prove that the leaves of the asymptotic foliation are the only large stable constant mean curvature spheres that enclose the center of $(M,g)$. This was shown previously under more restrictive decay assumptions and using a different method by S. Ma.

$\def\G{\overline{G}}$For a complex connected semisimple linear algebraic group $G$ of adjoint type and of rank $n$, De Concini and Procesi constructed its wonderful compactification $\G$, which is a smooth Fano $G \times G$-variety of Picard number $n$ enjoying many interesting properties. In this paper, it is shown that the wonderful compactification $\G$ is rigid under Fano deformation. Namely, for any regular family of Fano manifolds over a connected base, if one fiber is isomorphic to $\G$, then so are all other fibers. This answers a question raised by Bien and Brion in their work on the local rigidity of wonderful varieties.

Let $(A, \Delta)$ be a complex principally polarized abelian variety of dimension $g \geqslant 4$. Based on vanishing theorems, differentiation techniques and intersection theory, we show that whenever the theta divisor $\Delta$ is irreducible, its multiplicity at any point is at most $g-2$. This improves work of Kollár $\href{https://doi.org/10.1515/9781400864195}{[23]}$, Smith–Varley $\href{https://doi.org/ 10.1215/S0012-7094-96-08214-9}{[37]}$, and Ein–Lazarsfeld $\href{ https://www.jstor.org/stable/2152909}{[13]}$. We also introduce some new ideas to study the same type of questions for pluri-theta divisors.

The deformation of a variety $X$ to the normal cone of a subvariety $Y$ is a classical construction in algebraic geometry. In this paper we study the case when $(X, \omega)$ is a compact Kähler manifold and $Y$ is a submanifold. The deformation space $\mathcal{X}$ is fibered over $\mathbb{P}^1$ and all the fibers $X_\tau$ are isomorphic to $X$, except the zerofiber, which has the projective completion of the normal bundle $N_{Y|X}$ as one of its components. The first main result of this paper is that one can find Kähler forms on modifications of $\mathcal{X}$ which restricts to $\omega$ on $X_1$ and which makes the volume of the normal bundle in the zero-fiber come arbitrarily close to the volume of $X$. Phrased differently, we find Kähler deformations of $(X, \omega)$ such that almost all of the mass ends up in the normal bundle. The proof relies on a general result on the volume of big cohomology classes, which is the other main result of the paper. A $(1,1)$ cohomology class on a compact Kähler manifold $X$ is said to be big if it contains the sum of a Kähler form and a closed positive current. A quantative measure of bigness is provided by the volume function, and there is also a related notion of restricted volume along a submanifold. We prove that if $Y$ is a smooth hypersurface which intersects the Kähler locus of a big class $\alpha$ then up to a dimensional constant, the restricted volume of $\alpha$ along $Y$ is equal to the derivative of the volume at $\alpha$ in the direction of the cohomology class of $Y$. This generalizes the corresponding result on the volume of line bundles due to Boucksom–Favre–Jonsson and independently Lazarsfeld–Mustaţă.

Let $\Sigma$ be a compact oriented surface and $N$ a compact Kähler manifold with nonnegative holomorphic bisectional curvature. For a solution of harmonic map flow starting from an almost-holomorphic map $\Sigma \to N$ (in the energy sense), the limit at each singular time extends continuously over the bubble points and no necks appear.

We prove regularity, global existence, and convergence of Lagrangian mean curvature flows in the two-convex case (1.6). Such results were previously only known in the convex case, of which the current work represents a significant improvement. The proof relies on a newly discovered monotone quantity (2.6) that controls two-convexity. Through a unitary transformation, same result for the mean curvature flow of area-decreasing Lagrangian submanifolds (1.10) were established.

Gromov proved a cube inequality on the bound of distances between opposite faces of a cube equipped with a positive scalar curvature metric in dimension $\leq 8$ using a minimal surface method. He conjectured that the cube inequality also holds in dimension $\geq 9$. In this paper, we prove Gromov’s cube inequality in all dimensions with the optimal constant via a Dirac operator method.

Let $(X, p)$ be a Ricci limit space. We show that for any $\epsilon \gt 0$ and $x \in X$, there exists $r \lt \epsilon$, depending on $\epsilon$ and $x$, so that any loop in $B_r (x)$ is contractible in $B_\epsilon (x)$. In particular, $X$ is semi-locally simply connected. Then we show that the generalized Margulis lemma holds for Ricci limit spaces of $n$-manifolds.

$\def\Y{\mathscr{Y}}$In this paper, we introduce a new energy density function $\Y$ on the projective bundle $\mathbb{P}(T_M) \to M$ for a smooth map $f : (M,h) \to (N, g)$ between Riemannian manifolds\[\Y = g_{ij} f^i_\alpha f^j_\beta \dfrac{W^\alpha W^\beta}{\sum h_{\gamma\delta} W^\gamma W^\delta} \; \textrm{.}\]We establish new Hessian estimates to this energy density, which can be regarded as “average” versions of classical estimates. As applications, we obtain various new Liouville type theorems for holomorphic maps, harmonic maps and pluri-harmonic maps. For instance, we show that there is no non-constant holomorphic map from a compact Hermitian manifold with positive (resp. nonnegative) holomorphic sectional curvature to a Hermitian manifold with non-positive (resp. negative) holomorphic sectional curvature.