$\def\Vol{\operatorname{Vol}}$ Given a pair of metric tensors $g_1 \geq g_0$ on a Riemannian manifold, $M$, it is well known that $\Vol_1(M) \geq \Vol_0 (M)$. Furthermore one has rigidity: the volumes are equal if and only if the metric tensors are the same $g_1 = g_0$. Here we prove that if $g_j \geq g_0$ and $\Vol_j (M) \to \Vol_0 (M)$ then $(M, g_j)$ converge to $(M, g_0)$ in the volume preserving intrinsic at sense. Well known examples demonstrate that one need not obtain smooth, $C^0$, Lipschitz, or even Gromov–Hausdor convergence in this setting. Our theorem may also be applied as a tool towards proving other open conjectures concerning the geometric stability of a variety of rigidity theorems in Riemannian geometry. To complete our proof, we provide a novel way of estimating the intrinsic at distance between Riemannian manifolds which is interesting in its own right.

How large can be the width of Riemannian three-spheres of the same volume in the same conformal class? If a maximum value is attained, how does a maximising metric look like? What happens as the conformal class changes? In this paper, we investigate these and other related questions, focusing on the context of Simon–Smith min‑max theory.

We study how the existence of a negatively pinched Kähler metric on a domain in complex Euclidean space restricts the geometry of its boundary. In particular, we show that if a convex domain admits a complete Kähler metric, with pinched negative holomorphic bisectional curvature outside a compact set, then the boundary of the domain does not contain any complex subvariety of positive dimension. Moreover, if the boundary of the domain is smooth, then it is of finite type in the sense of D’Angelo. We also use curvature to provide a characterization of strong pseudoconvexity amongst convex domains. In particular, we show that a convex domain with $C^{2,\alpha}$ boundary is strongly pseudoconvex if and only if it admits a complete Kähler metric with sufficiently tight pinched negative holomorphic sectional curvature outside a compact set.

Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which relates, for a principal bundle $M \to M/H$, integrations of a Dirac structure on $M/H$ to $H$-admissible integrations of its pullback Dirac structure on $M$ by pre-symplectic groupoids. Our construction gives a distinguished class of explicit real or holomorphic pre-symplectic and symplectic groupoids over semi-simple Lie groups and some of their homogeneous spaces, including their symmetric spaces, conjugacy classes, and flag varieties. In a more general framework, we also show integrability of all homogeneous spaces of $LA^\vee$‑Lie groups in the sense of E. Meinrenken.

We develop a method of gluing the local mirrors and functors constructed from immersed Lagrangians in the same deformation class. As a result, we obtain a canonical mirror functor to the glued category. We apply the method to construct the mirrors of punctured Riemann surfaces and show that our functor derives homological mirror symmetry.

We prove that the Hilbert scheme which parametrises bitangent lines to a general quartic surface is a smooth regular surface with no rational curves and with very ample canonical divisor. We also prove that it is a counterexample to infinitesimal Torelli and that its infinitesimal deformation space has dimension $20$.

The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkähler 4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian isotopy classes of Lagrangians, and the Thomas-Yau conjecture on longtime existence of the Lagrangian mean curvature ow. We also make observations concerning closed geodesics, curve shortening flow and minimal surfaces.

This paper settles the question of injectivity of the non-Abelian X-ray transform on simple surfaces for the general linear group of invertible complex matrices. The main idea is to use a factorization theorem for Loop Groups to reduce to the setting of the unitary group, where energy methods and scalar holomorphic integrating factors can be used. We also show that our main theorem extends to cover the case of an arbitrary Lie group.

$\def\mth{ m\textrm{-th}}$ Let $X$ be a compact connected CR manifold with a transversal CR $S^1$-action of real dimension $2n-1$, which is only assumed to be weakly pseudoconvex. Let $\square_b$ be the $\overline{\partial}_b$-Laplacian, with respect to a $T$-rigid Hermitian metric (see Definition 3.2 of $T$-rigid Hermitian metric). Eigenvalue estimate of $\square_b$ is a fundamental issue both in CR geometry and analysis. In this paper, we are able to obtain a sharp estimate of the number of eigenvalues smaller than or equal to $\lambda$ of $\square_b$ acting on the $\mth$ Fourier components of smooth $(n-1,q)$-forms on $X$, where $m \in \mathbb{Z}_{+}$ and $q = 0, 1, \dotsc , n-1$. Here the sharp means the growth order with respect to $m$ is sharp. In particular, when $\lambda = 0$, we obtain the asymptotic estimate of the growth for $\mth$ Fourier components $H^{n-1,q}_{b,m} (X)$ of $H^{n-1,q}_b (X)$ as $m \to + \infty$. Furthermore, we establish a Serre type duality theorem for Fourier components of Kohn–Rossi cohomology which is of independent interest. As a byproduct, the asymptotic growth of the dimensions of the Fourier components $H^{0,q}_{b,-m} (X)$ for $m \in \mathbb{Z}_{+}$ is established. We also give applications of our main results, including Morse type inequalities, asymptotic Riemann–Roch type theorem, Grauert–Riemenschneider type criterion, and an orbifold version of our main results which provides an answer towards a folklore open problem informed to us by Hsiao.