The aim of this paper is to establish the correspondence between the twisted localised Pestov identity on the unit tangent bundle of a Riemannian manifold and the Weitzenböck identity for twisted symmetric tensors on the manifold.

In this paper, we investigate hypersurfaces of $\mathbb{S}^2\times \mathbb{S}^2$ and $\mathbb{H}^2\times \mathbb{H}^2$ with recurrent Ricci tensor. As the main result, we prove that a hypersurface in $\mathbb{S}^2\times \mathbb{S}^2$ (resp. $\mathbb{H}^2\times \mathbb{H}^2$) with recurrent Ricci tensor is either an open part of $\Gamma \times \mathbb{S}^2$ (resp. $\Gamma \times \mathbb{H}^2$) for a curve $\Gamma$ in $\mathbb{S}^2$ (resp. $\mathbb{H}^2$), or a hypersurface with constant sectional curvature. The latter has been classified by H. Li, L. Vrancken, X. Wang, and Z. Yao very recently.

We explore the regularity theory of optimal transport maps for costs satisfying a Ma–Trudinger–Wang condition, by viewing the graphs of the transport maps as maximal Lagrangian surfaces with respect to an appropriate pseudo-Riemannian metric on the product space. We recover the local regularity theory in two-dimensional manifolds.

We show that the fundamental groups of smooth $4$-manifolds that admit geometric decompositions in the sense of Thurston have asymptotic dimension at most four, and equal to four when aspherical. We also show that closed $3$-manifold groups have asymptotic dimension at most three. Our proof method yields that the asymptotic dimension of closed $3$-dimensional Alexandrov spaces is at most three. Thus, we obtain that the Novikov conjecture holds for closed $4$-manifolds with such a geometric decomposition and for closed $3$-dimensional Alexandrov spaces. Consequences of these results include a vanishing result for the Yamabe invariant of certain $0$-surgered geometric $4$-manifolds and the existence of zero in the spectrum of aspherical smooth $4$-manifolds with a geometric decomposition.

Let $X$ be a compact Riemann surface. Let $(E,\theta )$ be a stable Higgs bundle of degree $0$ on $X$. Let $h_{\det (E)}$ denote a flat metric of the determinant bundle $\det (E)$. For any $t\gt 0$, there exists a unique harmonic metric $h_t$ of $(E,t\theta )$ such that $\det (h_t)=h_{\det (E)}$. We prove that if the Higgs bundle is induced by a line bundle on the normalization of the spectral curve, then the sequence $h_t$ is convergent to the naturally defined decoupled harmonic metric at the speed of the exponential order. We also obtain a uniform convergence for such a family of Higgs bundles.

In this paper, we prove the non-existence of Codazzi and totally umbilical hypersurfaces, especially totally geodesic hypersurfaces, in the $4$-dimensional model space $\mathrm {Sol}_1^4$.

In this paper we discuss three distance functions on the set of convex bodies. In particular we study the convergence of Delzant polytopes, which are fundamental objects in symplectic toric geometry. By using these observations, we derive some convergence theorems for symplectic toric manifolds with respect to the Gromov–Hausdorff distance.

We provide two constructions of Gaussian random holomorphic sections of a Hermitian holomorphic line bundle $(L,h_{L})$ on a Hermitian complex manifold $(X,\Theta )$, that are particularly interesting in the case where the space of $\mathcal {L}^2$-holomorphic sections $H^{0}_{(2)}(X,L)$ is infinite dimensional. We first provide a general construction of Gaussian random holomorphic sections of L, which, if $H^{0}_{(2)}(X,L)$ is infinite dimensional, are almost never $\mathcal {L}^2$-integrable on X. The second construction combines the abstract Wiener space theory with the Berezin–Toeplitz quantization and yields a Gaussian ensemble of random $\mathcal {L}^2$-holomorphic sections. Furthermore, we study their random zeros in the context of semiclassical limits, including their distributions, large deviation estimates, local fluctuations and hole probabilities.

In this article, we study the behavior of complete two-sided hypersurfaces immersed in the hyperbolic space $\mathbb H^{n+1}$. Initially, we introduce the concept of the linearized curvature function $\mathcal {F}_{r,s}$ of a two-sided hypersurface, its associated modified Newton transformation $\mathcal {P}_{r,s}$ and its naturally attached differential operator $\mathcal {L}_{r,s}$. Then, we obtain two formulas for differential operator $\mathcal {L}_{r,s}$ acting on the height function of a two-sided hypersurface and, for the case where their support functions are related by a negative constant, we derive two new formulas for the Newton transformation $P_{r}$ and the modified Newton transformation $\mathcal {P}_{r,s}$ acting on a gradient of the height function. Finally, these formulas, jointly with suitable maximum principles, enable us to establish our rigidity and nonexistence results concerning complete two-sided hypersurfaces in $\mathbb H^{n+1}$.

We prove that the only Bott manifolds such that the Futaki invariant vanishes for any Kähler class are isomorphic to the products of the projective lines.

We prove two sharp anisotropic weighted geometric inequalities that hold for star-shaped and F-mean convex hypersurfaces in $\mathbb{R}^{n+1}$, which involve the anisotropic p-momentum, the anisotropic perimeter, and the volume of the region enclosed by the hypersurface. We also consider their quantitative versions characterized by asymmetry index and the Hausdorff distance between the hypersurface and a rescaled Wulff shape. As a corollary, we obtain the stability of the Weinstock inequality for the first non-zero Steklov eigenvalue for star-shaped and strictly mean convex domains.

We show that March’s criterion for the existence of a bounded nonconstant harmonic function on a weak model (that is, $\mathbb {R}^n$ with a rotationally symmetric metric) is also a necessary and sufficient condition for the solvability of the Dirichlet problem at infinity on a family of metrics that generalise metrics with rotational symmetry on $\mathbb {R}^n$. When the Dirichlet problem at infinity is not solvable, we prove some quantitative estimates on how fast a nonconstant harmonic function must grow.

We express the total space of a principal circle bundle over a connected sum of two manifolds in terms of the total spaces of circle bundles over each summand, provided certain conditions hold. We then apply this result to provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres and obtain a topological classification of closed, simply connected manifolds with a free cohomogeneity-four torus action. As a corollary, we obtain infinitely many manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions.

We prove a synthetic Bonnet–Myers rigidity theorem for globally hyperbolic Lorentzian length spaces with global curvature bounded below by K < 0 and an open distance realizer of length $L=\frac{\pi}{\sqrt{|K|}}$: It states that the space necessarily is a warped product with warping function $\cos: (-\frac{\pi}{2},\frac{\pi}{2})\to\mathbb{R}_+$. From this, one also sees that a globally hyperbolic spacetime with curvature bounded above by K < 0 and an open distance realizer of length $L=\frac{\pi}{\sqrt{|K|}}$ is a warped product with warping function cos.

Let $(\tau , V_{\tau })$ be a finite dimensional representation of a maximal compact subgroup K of a connected non-compact semisimple Lie group G, and let $\Gamma $ be a uniform torsion-free lattice in G. We obtain an infinitesimal version of the celebrated Matsushima–Murakami formula, which relates the dimension of the space of automorphic forms associated to $\tau $ and multiplicities of irreducible $\tau ^\vee $-spherical spectra in $L^2(\Gamma \backslash G)$. This result gives a promising tool to study the joint spectra of all central operators on the homogenous bundle associated to the locally symmetric space and hence its infinitesimal $\tau $-isospectrality. Along with this, we prove that the almost equality of $\tau $-spherical spectra of two lattices assures the equality of their $\tau $-spherical spectra.

Quasigeodesic behavior of flow lines is a very useful property in the study of Anosov flows. Not every Anosov flow in dimension three is quasigeodesic. In fact, until recently, up to orbit equivalence, the only previously known examples of quasigeodesic Anosov flows were suspension flows. In a recent article, the second author proved that an Anosov flow on a hyperbolic 3-manifold is quasigeodesic if and only if it is non-$\mathbb {R}$-covered, and this result completes the classification of quasigeodesic Anosov flows on hyperbolic 3-manifolds. In this article, we prove that a new class of examples of Anosov flows are quasigeodesic. These are the first examples of quasigeodesic Anosov flows on 3-manifolds that are neither Seifert, nor solvable, nor hyperbolic. In general, it is very hard to show that a given flow is quasigeodesic and, in this article, we provide a new method to prove that an Anosov flow is quasigeodesic.

In this paper, we prove an optimal isoperimetric inequality for spacelike, compact, star-shaped, and $2$-convex hypersurfaces in de Sitter space.

Following the work of Mazzeo–Swoboda–Weiß–Witt [Duke Math. J. 165 (2016), 2227–2271] and Mochizuki [J. Topol. 9 (2016), 1021–1073], there is a map $\overline{\Xi }$ between the algebraic compactification of the Dolbeault moduli space of ${\rm SL}(2,\mathbb{C})$ Higgs bundles on a smooth projective curve coming from the $\mathbb{C}^\ast$ action and the analytic compactification of Hitchin’s moduli space of solutions to the $\mathsf{SU}(2)$ self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ‘limiting configurations’. This map extends the classical Kobayashi–Hitchin correspondence. The main result that this article will show is that $\overline{\Xi }$ fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration.

The fixed points of the generalized Ricci flow are the Bismut Ricci flat (BRF) metrics, i.e., a generalized metric (g, H) on a manifold M, where g is a Riemannian metric and H a closed 3-form, such that H is g-harmonic and $\operatorname{Rc}(g)=\tfrac{1}{4} H_g^2$. Given two standard Einstein homogeneous spaces $G_i/K$, where each Gi is a compact simple Lie group and K is a closed subgroup of them holding some extra assumption, we consider $M=G_1\times G_2/\Delta K$. Recently, Lauret and Will proved the existence of a BRF metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on M among a subset of G-invariant metrics and, if $G_1=G_2$, then it is globally stable.

The famous Cheng-Shen’s conjecture in Riemann-Finsler geometry claims that every n-dimensional closed W-quadratic Randers manifold is a Berwald manifold. In this paper, first we study the Riemann and Ricci curvatures of homogeneous Finsler manifolds and obtain some rigidity theorems. Then, by using this investigation, we construct a family of W-quadratic Randers metrics which are not R-quadratic nor strongly Ricci-quadratic.