We prove a flatness result for entire nonlocal minimal graphs having some partial derivatives bounded from either above or below. This result generalizes fractional versions of classical theorems due to Bernstein and Moser. Our arguments rely on a general splitting result for blow-downs of nonlocal minimal graphs. Employing similar ideas, we establish that entire nonlocal minimal graphs bounded on

We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner–Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first result is an asymptotic solution to all orders. This involves log terms. We show that

In this paper, we construct Delaunay type constant mean curvature surfaces along a nondegenerate closed geodesic in a 3‑dimensional Riemannian manifold.

We provide a suitable axiomatic framework for differential cohomology in the relative case and we deduce the corresponding long exact sequences. We also construct the relative version of the generalized Cheeger–Simons characters and we define the integration map when the fibre has a boundary, generalizing to any cohomology theory the results of [13] (F. Ferrari Ruffino, “Relative Deligne cohomology

The Bochner tensor is the Kähler analogue of the conformal Weyl tensor. In this article, we derive local (i.e., in a neighbourhood of almost every point) normal forms for a (pseudo-)Kähler manifold with vanishing Bochner tensor. The description is pined down to a new class of symmetric spaces which we describe in terms of their curvature operators. We also give a local description of weakly Bochner-flat

We prove that Maxwell fields of asymptotically flat solutions of the Einstein–Maxwell equations inherit the stationarity of the metric.

We show that the conformal structure for the Riemannian analogues of Kerr black-hole metrics can be given an ambitoric structure. We then discuss the properties of the moment maps. In particular, we observe that the moment map image is not locally convex near the singularity corresponding to the ring singularity in the interior of the black hole. We then proceed to classify regular ambitoric $4$-orbifolds

We investigate some differential geometric aspects of the space of Bridgeland stability conditions on a Calabi–Yau triangulated category The aim is to give a provisional definition of Weil–Petersson geometry on the stringy Kähler moduli space. We study in detail a few basic examples to support our proposal. In particular, we identify our Weil–Petersson metric with the Bergman metric on a Siegel modular

We show that some embedded standard $13$‑spheres in Shimada’s exotic $15$‑spheres have $\mathbb{Z}_2$ quotient spaces, $P^{13}$s, that are fake real $13$‑dimensional projective spaces, i.e., they are homotopy equivalent, but not diffeomorphic to the standard $\mathbb{R}\mathbf{P}^{13}$. As observed by F. Wilhelm and the second named author in [RW], the Davis $\mathsf{SO}(2) \times \mathsf{G}_2$ actions

We show that, among free boundary minimal surfaces in the unit ball in the three-dimensional Euclidean space, the flat equatorial disk and the critical catenoid are characterised by a pinching condition on the length of their second fundamental form.

We study the index of the APS boundary value problem for a strongly Callias-type operator $\mathcal{D}$ on a complete Riemannian manifold $M$. We use this index to define the relative $\eta$-invariant $\eta (\mathcal{A}_1 , \mathcal{A}_0)$ of two strongly Callias-type operators, which are equal outside of a compact set. Even though in our situation the $\eta$-invariants of $\mathcal{A}_1$ and $\mathcal{A}_0$

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete

Given an asymptotically hyperbolic manifold with minimal conformal infinity, we construct blowing-up solutions for linear perturbations of the fractional Yamabe problem on the conformal infinity provided that either the trace-free part of the second fundamental form or the covariant normal derivative of the normal component of the Ricci tensor on the conformal infinity is non-trivial.

This article is devoted to the study of prime alternating +achiral knots. In the case of arborescent knots, we prove in +AAA Visibility Theorem 5.1, that the symmetry is visible on a certain projection (not necessarily minimal) and that it is realised by a homeomorphism of order $4$. In the general case (arborescent or not), if the prime alternating knot has no minimal projection on which +achirality

For every genus $g \geqslant 2$, we construct an infinite family of strongly quasipositive fibred knots $K_n$ having the same Seifert form as the torus knot $T(2, 2g + 1)$. In particular, their homological monodromies agree and their signatures and four-genera are maximal: $\lvert \sigma (K_n) \rvert = 2g_4 (K_n) = 2g$. On the other hand, the geometric stretching factors are pairwise distinct and the

In this paper, we study the Gauss map of a free boundary minimal surface. The main theorem asserts that if components of the Gauss map are eigenfunctions of the Jacobi–Steklov operator, then the surface must be rotationally symmetric.

In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota’s work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative curvature operator, finite asymptotic entropy implies $\kappa$‑noncollapsing on all

Let $G$ be a reductive affine algebraic group defined over $\mathbb{C}$, and let $\nabla_0$ be a meromorphic $G$-connection on a holomorphic principal $G$-bundle $E_0$, over a smooth complex projective curve $X_0$, with polar locus $P_0 \subset X_0$. We assume that $\nabla_0$ is irreducible in the sense that it does not factor through some proper parabolic subgroup of $G$. We consider the universal

We provide a geometric framework for the construction of non-vacuum black holes whose metrics are stationary and axisymmetric. Under suitable assumptions we show that the Einstein equations reduce to an Einstein-harmonic map type system and analyze the compatibility of the resulting equations. This framework is fundamental to our construction [3] of metric-stationary axisymmetric bifurcations of Kerr

We consider sub-Riemannian spaces admitting an isometry group that is maximal in the sense that any linear isometry between the horizontal tangent spaces is realized by a global isometry. We will show that these spaces have a canonical choice of partial connection on their horizontal bundle, which is determined by isometries and generalizes the Levi–Civita connection for the special case of Riemannian

We endow each closed, orientable Alexandrov space $(X, d)$ with an integral current $T$ of weight equal to $1 , \partial T = 0$ and $\operatorname{set}(T) = X$, in other words, we prove that $(X, d, T)$ is an integral current space with no boundary. Combining this result with a result of Li and Perales, we show that non-collapsing sequences of these spaces with uniform lower curvature and diameter

We provide regularity results for CR-maps from a real hypersurface in $n$-dimensional complex space to a Levi-degenerate target hypersurface in a $n + 1$-dimensional space. We address both the real-analytic and the smooth case. Our results allow immediate applications to the study of proper holomorphic maps between Bounded Symmetric Domains.

We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in $\mathbb{R}^n$. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the surface.We establish longtime-existence of the flow and investigate the projection of the flowing surface onto $\mathbb{R}^n$, the shadow of the flow. This moving

The conformal method is a technique for finding Cauchy data in general relativity solving the Einstein constraint equations, and its parameters include a conformal class, a conformal momentum (as measured by a densitized lapse), and a mean curvature. Although the conformal method is successful in generating constant mean curvature (CMC) solutions of the constraint equations, it is unknown how well

We show that for very general hypersurfaces in odd-dimensional simplicial projective toric varieties satisfying an effective combinatorial property the Hodge conjecture holds. This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.

This note adapts the sophisticated Richberg technique for approximation in pluripotential theory to the $F$-potential theory associated to a general nonlinear convex subequation $F \subset J^2 (X)$ on a manifold $X$. The main theorem is the following “local to global” result. Suppose $u$ is a continuous strictly $F$-subharmonic function such that each point $x \in X$ has a fundamental neighborhood

A smooth curve $\gamma$ in $\mathbb{R}^{n+1,n}$ is isotropic if $\gamma , \gamma_x, \dotsc , \gamma^{(2n)}_x$ are linearly independent and the span of $\gamma , \gamma_x, \dotsc , \gamma^{(n−1)}_x$ is isotropic. We construct two hierarchies of isotropic curve flows on $\mathbb{R}^{n+1,n}$, whose differential invariants are solutions of Drinfeld–Sokolov’s KdV type soliton hierarchies associated to the

We study analytic properties of harmonic maps from Riemannian polyhedra into $\operatorname{CAT}(\kappa)$ spaces for $\kappa \in {\lbrace 0, 1 \rbrace}$. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into $\operatorname{CAT}(\kappa)$ spaces. We compute a target variation formula that captures the curvature bound in the target, and use

We provide sufficient conditions on an initial curve for the area preserving and the length preserving curvature flows of curves in a plane, to develop a singularity at some finite time or converge to an $m$-fold circle as time goes to infinity. For the area-preserving flow, the positivity of the enclosed algebraic area determines whether the curvature blows up in finite time or not, while for the

We examine Higgs bundles for non-compact real forms of $SO(4,\mathbb{C})$ and the isogenous complex group $SL(2,\mathbb{C}) \times SL(2,\mathbb{C})$. This involves a study of non-regular fibers in the corresponding Hitchin fibrations and provides interesting examples of non-abelian spectral data.

We describe the invariant metrics on real flag manifolds and classify those with the following property: every geodesic is the orbit of a one-parameter subgroup. Such a metric is called g.o. (geodesic orbit). In contrast to the complex case, on real flag manifolds the isotropy representation can have equivalent submodules, which makes invariant metrics depend on more parameters and allows us to find

For any elliptic K3 surface $\mathfrak{F} : \mathcal{K} \to \mathbb{P}^1$, we construct a family of collapsing Ricci-flat Kähler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov–Hausdorff limit to $\mathbb{P}^1$ equipped with the McLean metric. There are well-known examples of this type of collapsing, but the key point of our construction is that we can

This article considers the quasi-local conserved quantities with respect to a reference spacetime with a cosmological constant. We follow the approach developed by the authors in [7, 26, 27] and define the quasi-local energy as differences of surface Hamiltonians. The ground state for the gravitational energy is taken to be a reference configuration in the de Sitter (dS) or Anti‑de Sitter (AdS) spacetime

We prove nonlinear stability for a large class of solutions to the Einstein equations with a positive cosmological constant and compact spatial topology in arbitrary dimensions, where the spatial metric is Einstein with either positive or negative Einstein constant. The proof uses the CMC Einstein flow and stability follows by an energy argument.We prove in addition that the development of non-CMC

We show existence and we give a geometric characterization of isoperimetric regions for large volumes, in $C^2$-locally asymptotically Euclidean Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends. This work extends previous results contained in [EM13b], [EM13a], and [BE13]. Moreover strengthening a little bit the speed of convergence to the Schwarzschild metric we

The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kähler case. Our main question is the existence of almost Kähler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in

A slope $p/q$ is a characterizing slope for a knot $K$ in $S^3$ if the oriented homeomorphism type of $p/q$-surgery on $K$ determines $K$ uniquely. We show that for each torus knot its set of characterizing slopes contains all but finitely many non-integer slopes. This generalizes work of Ni and Zhang who established such a result for $T_{5,2}$. Along the way we show that if two knots $K$ and $K^\prime$

We consider solutions $(M, g(t)), 0 \leq t \lt T$, to Ricci flow on compact, connected four dimensional manifolds without boundary. We assume that the scalar curvature is bounded uniformly, and that $T \lt \infty$. In this case, we show that the metric space $(M, d(t))$ associated to $(M, g(t))$ converges uniformly in the $C^0$ sense to $(X, d)$, as $t \nearrow T$, where $(X, d)$ is a $C^0$ Riemannian

Seven years after the publication of “Smooth convergence away from singular sets” [Communications in Analysis and Geometry 21 (2013), no. 1, 39‑104], Brian Allen discovered a counter example to the published statement of Theorem 1.3. Note that Theorem 4.6 (which is the key theorem cited in other papers) remains correct. We have added an hypothesis to correct the statement of Theorem 1.3 and its consequences

In this paper, we introduce a geometric flow for Kähler metrics $\omega_t$ coupled with closed $(1,1)$‑forms $\alpha_t$ on a compact Kähler manifold, whose stationary solution is a constant scalar curvature Kähler (cscK) metric, coupled with a harmonic $(1,1)$‑form. We establish the long-time existence, i.e., assuming the initial $(1,1)$‑form $\alpha$ is nonnegative, then the flow exists as long as

We consider a variational problem of $p$-elastic curves in two-dimensional sphere. We give its first variation formula, and in two-dimensional sphere, we give a realization of a solution which satisfies that the first variation formula is zero. We also show the existence of a flat-core, closed $p$-elastic curve.

We consider smooth solutions (M,g(t)), 0 <= t

This article considers the quasi-local conserved quantities with respect to a reference spacetime with a cosmological constant. We follow the approach developed by the authors in [25,26,7] and define the quasi-local energy as differences of surface Hamiltonians. The ground state for the gravitational energy is taken to be a reference configuration in the de Sitter (dS) or Anti-de Sitter (AdS) spacetime

In this note we show how a generalized Pohozaev-Schoen identity due to Gover and Orsted \cite{GO} can be used to obtain some rigidity results for $V$-static manifolds and generalized solitons. We also obtain an Alexandrov type result for certain hypersurfaces in Einstein manifolds.

The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the K\"ahler case. Our main question is the existence of almost K\"ahler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and

A slope $p/q$ is a characterizing slope for a knot $K$ in $S^3$ if the oriented homeomorphism type of $p/q$-surgery on $K$ determines $K$ uniquely. We show that for each torus knot its set of characterizing slopes contains all but finitely many non-integer slopes. This generalizes work of Ni and Zhang who established such a result for $T_{5,2}$. Along the way we show that if two knots $K$ and $K'$

We use the LMO invariant to find constraints for a knot to admit a purely or reflectively cosmetic surgery. We also get a constraint for knots to admit a Lens space surgery, and some information for characterizing slopes.

We prove nonlinear stability for a large class of solutions to the Einstein equations with a positive cosmological constant and compact spatial topology in arbitrary dimensions, where the spatial metric is Einstein with either positive or negative Einstein constant. The proof uses the CMC Einstein flow and stability follows by an energy argument. We prove in addition that the development of non-CMC

We define the {\it Wirtinger number} of a link, an invariant closely related to the meridional rank. The Wirtinger number is the minimum number of generators of the fundamental group of the link complement over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a link equals its bridge number. This equality

We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has $L^2$-bounded second fundamental form and satisfies a weak power growth on the area. We give the precise asymptotic behavior of an end of such a surface. This asymptotic

We generalize the ancient solutions of the Ricci flow on certain principal ${\rm SO}(3)$ bundles over compact quaternionic K\"ahler manifolds constructed by Bakas, Kong, and Ni to certain $RP^3$ fibre bundles over a product of two compact quaternionic K\"ahler manifolds. The ancient solutions are of Type I, $\kappa$-noncollapsed, and have positive Ricci curvature. Analogous solutions on even-dimensional

This paper looks at the splitting problem for globally hyperbolic spacetimes with timelike Ricci curvature bounded below containing a (spacelike, acausal, future causally complete) hypersurface with mean curvature bounded from above. For such spacetimes we show a splitting theorem under the assumption of either the existence of a ray of maximal length or a maximality condition on the volume of Lorentzian

In this paper, we investigate under which conditions a differential inequality F on a manifold X satisfies a Liouville-type property. This question, arising in nonlinear potential theory, is related to the validity of an appropriate maximum principle at infinity for F (called here the Ahlfors property) which for instance includes, for suitable F, the classical Ekeland and Omori-Yau principles, their

We prove dynamical stability and instability theorems for compact Einstein metrics under the Ricci flow. We give a nearly complete charactarization of dynamical stability and instability in terms of the conformal Yamabe invariant and the Laplace spectrum. In particular, we prove dynamical stability of some classes of Einstein manifolds for which it was previously not known. Additionally, we show that

In this paper, we compute the Morse index for a free boundary minimal submanifold from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. As an application, we show that the Morse index of a free boundary minimal annulus is equal to 4 if and only if it is the critical

Finsler metrics of scalar flag curvature play an important role to show the complexity and richness of general Finsler metrics. In this paper, on an $n$-dimensional manifold $M$ we study the Finsler metric $F=F(x,y)$ of scalar flag curvature ${\bf K} = {\bf K}(x,y)$ and discover some equations ${\bf K}$ should be satisfied. As an application, we mainly study the metric $F$ of weakly isotropic flag

In this paper, we give an almost complete classification of toric surface codes of dimension less than or equal to 7, according to monomially equivalence. This is a natural extension of our previous work [YZ], [LYZZ]. More pairs of monomially equivalent toric codes constructed from non-equivalent lattice polytopes are discovered. A new phenomenon appears, that is, the monomially non-equivalence of

We solve the classical problem of Plateau in every metric space which is $1$-complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual Banach spaces, some non-dual Banach spaces such as $L^1$, all Hadamard spaces, and many more. Our results generalize corresponding results of Lytchak and the second

We show existence and caracterization of isoperimetric regions for large volumes, in C 0 -locally asymptotically Euclidean Riemannian manifolds with a finite number of C 0 -asymptotically Schwarzschild ends. Extending previous results of (EM13b) and in (EM13a). Such a results is of interest for mathematical general relativity.

Counter-examples to the famous conjecture of Caratheodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean 3-space. In particular, Riemannian metrics with a smooth strictly convex 2-sphere containing a single umbilic point are constructed explicitly, in contradiction with any direct