We study the K-stability of blow-ups of the weighted projective plane ${\mathbb P}(1,1,n)$, where n is a positive integer.

We prove that the proximal unit normal bundle of the subgraph of a $W^{2,n} $-function carries a natural structure of Legendrian cycle. This result is used to obtain an Alexandrov-type sphere theorem for hypersurfaces in $ \mathbf{R}^{n+1} $, which are locally graphs of arbitrary $W^{2,n} $-functions. We also extend the classical umbilicality theorem to $ W^{2,1} $-graphs, under the Lusin (N) condition for the graph map.

Given a simply connected manifold M, we completely determine which rational monomial Pontryagin numbers are attained by fiber homotopy trivial M-bundles over the k-sphere, provided that k is small compared to the dimension and the connectivity of M. Furthermore, we study the vector space of rational cobordism classes represented by such bundles. We give upper and lower bounds on its dimension, and we construct manifolds for which the lower bound is attained. Our proofs are based on the classical approach to studying diffeomorphism groups via block bundles and surgery theory, and we make use of ideas developed by Krannich–Kupers–Randal-Williams.

As an application, we show the existence of elements of infinite order in the homotopy groups of the spaces of positive Ricci and positive sectional curvature, provided that M is $\operatorname {Spin}$, has a nontrivial rational Pontryagin class and admits such a metric. This is done by constructing M-bundles over spheres with nonvanishing ${\hat {\mathcal {A}}}$-genus. Furthermore, we give a vanishing theorem for generalized Morita–Miller–Mumford classes for fiber homotopy trivial bundles over spheres.

In the appendix coauthored by Jens Reinhold, we investigate which classes of the rational oriented cobordism ring contain an element that fibers over a sphere of a given dimension.

We study the geometry induced on the local orbit spaces of Killing vector fields on (Riemannian) $G$-manifolds, with an emphasis on the cases $G=\textrm {Spin}(7)$ and $G=G_2$. Along the way, we classify the harmonic morphisms with one-dimensional fibres from $G_2$-manifolds to Einstein manifolds.

We establish a one-to-one correspondence between Kähler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non-linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to $\Lambda^3(\mathcal{T})$, where ${\mathcal{T}}$ is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms.

Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a Kähler metric. In particular, we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing–Yano tensor, which is a necessary condition for a metric to be conformal to Kähler. This gives an invariant characterization of algebraically special Riemannian metrics of type D in dimensions higher than four.

The 3-dimensional Heisenberg group can be equipped with three different types of left-invariant Lorentzian metric, according to whether the center of the Lie algebra is spacelike, timelike or null. Using the second of these types, we study spacelike surfaces of mean curvature zero. These surfaces with singularities are associated with harmonic maps into the 2-sphere. We show that the generic singularities are cuspidal edge, swallowtail and cuspidal cross-cap. We also give the loop group construction for these surfaces, and the criteria on the loop group potentials for the different generic singularities. Lastly, we solve the Cauchy problem for harmonic maps into the 2-sphere using loop groups, and use this to give a geometric characterisation of the singularities. We use these results to prove that a regular spacelike maximal disc with null boundary must have at least two cuspidal cross-cap singularities on the boundary.

We investigate the Lorentzian analogues of Riemannian Bianchi–Cartan–Vranceanu spaces. We provide their general description and emphasize their role in the classification of three-dimensional homogeneous Lorentzian manifolds with a four-dimensional isometry group. We then illustrate their geometric properties (with particular regard to curvature, Killing vector fields and their description as Lorentzian Lie groups) and we study several relevant classes of surfaces (parallel, totally umbilical, minimal, constant mean curvature) in these homogeneous Lorentzian three-manifolds.

We discuss the Singer conjecture and Gromov–Lück inequality $\chi\geq |\sigma|$ for aspherical complex surfaces. We give a proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group that does not rely on Gromov’s Kähler groups theory. Without the residually finiteness assumption, we observe that this conjecture can be proven for all aspherical complex surfaces except possibly those in Class $\mathrm{VII}_0^+$ (a positive answer to the global spherical shell conjecture would rule out the existence of aspherical surfaces in this class). We also sharpen the Gromov-Lück inequality for aspherical complex surfaces that are not in Class $\mathrm{VII}_0^+$. This is achieved by connecting the circle of ideas of the Singer conjecture with the study of Reid’s conjecture.

Special Ricci–Hessian equations on Kähler manifolds $(M,g)$, as defined by Maschler [‘Special Kähler–Ricci potentials and Ricci solitons’, Ann. Global Anal. Geom. 34 (2008), 367–380], involve functions $\tau $ on M and state that, for some function $\alpha $ of the real variable $\tau\kern-0.8pt $, the sum of $\alpha \nabla d\tau\kern-0.8pt $ and the Ricci tensor equals a functional multiple of the metric g, while $\alpha \nabla d\tau\kern-0.8pt $ itself is assumed to be nonzero almost everywhere. Three well-known obvious ‘standard’ cases are provided by (non-Einstein) gradient Kähler–Ricci solitons, conformally-Einstein Kähler metrics, and special Kähler–Ricci potentials. We show that, outside of these three cases, such an equation can only occur in complex dimension two and, at generic points, it must then represent one of three types, for which, up to normalizations, $\alpha =2\cot \tau\kern-0.8pt $, $\alpha =2\coth \tau\kern-0.8pt $, or $\alpha =2\tanh \tau\kern-0.8pt $. We also use the Cartan–Kähler theorem to prove that these three types are actually realized in a ‘nonstandard’ way.

Let M be a smooth closed oriented surface. Gaussian thermostats on M correspond to the geodesic flows arising from metric connections, including those with non-zero torsion. These flows may not preserve any absolutely continuous measure. We prove that if two Gaussian thermostats on M with negative thermostat curvature are related by a smooth orbit equivalence isotopic to the identity, then the two background metrics are conformally equivalent via a smooth diffeomorphism of M isotopic to the identity. We also give a relationship between the thermostat forms themselves. Finally, we prove the same result for Anosov magnetic flows.

We prove a strong Frankel theorem for mean curvature flow shrinkers in all dimensions: Any two shrinkers in a sufficiently large ball must intersect. In particular, the shrinker itself must be connected in all large balls. The key to the proof is a strong Bernstein theorem for incomplete stable Gaussian surfaces.

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 $\mathbb H^{n+1}$.