The classical Patterson–Walker construction of a split-signature (pseudo-)Riemannian structure from a given torsion-free affine connection is generalized to a construction of a split-signature conformal structure from a given projective class of connections. A characterization of the induced structures is obtained. We achieve a complete description of Einstein metrics in the conformal class formed

In this article we prove that, if $X$ is a smooth $4$-manifold containing an embedded double node neighborhood, all knot surgery $4$-manifolds $X_K$ are mutually diffeomorphic to each other after a connected sum with $\mathbb{CP}^2$. Hence, by applying to the simply connected elliptic surface $E(n)$, we also show that every knot surgery $4$-manifold $E(n)_K$ is almost completely decomposable.

Let $(M, g)$ be a pseudo-Riemannian manifold of arbitrary dimension and signature. We prove that there exist mutually quasi-inverse equivalences between the groupoid of weakly faithful real pinor bundles on $(M, g)$ and the groupoid of weakly faithful real Lipschitz structures on $(M, g)$, from which follows that every bundle of weakly faithful real Clifford modules is associated to a real Lipschitz

Let $\alpha \in [0, 1)$ and $\Omega$ be an open connected subset of $\mathbb{R}^{n \geq 2}$. This paper shows that the $Q_{\alpha}$-restriction problem $Q_{\alpha} \vert {}_{\Omega} = \mathscr{Q}_{\alpha} (\Omega)$ is solvable if and only if $\Omega$ is an Ahlfors $n$-regular domain; i.e., $\operatorname{vol} \bigl ( B(x, r) \cap \Omega \bigr ) \gtrsim r^n$ for any Euclidean ball $B(x, r)$ with center

In this paper we study $F$-manifolds equipped with multiple flat connections and multiple $F$-products, that are required to be compatible in a suitable sense. Multi-flat $F$-manifolds are the analogue for $F$-manifolds of Frobenius manifolds with multi-Hamiltonian structures. In the semisimple case, we show that a necessary condition for the existence of such multiple flat connections can be expressed

In every point of a Kähler manifold there exist special holomorphic coordinates well adapted to the underlying geometry. Comparing these Kähler normal coordinates with the Riemannian normal coordinates defined via the exponential map we prove that their difference is a universal power series in the curvature tensor and its iterated covariant derivatives and devise an algorithm to calculate this power

In this paper, we will consider generalised eigenfunctions of the Laplacian on some surfaces of infinite area. We will be interested in lower bounds on the number of nodal domains of such eigenfunctions which are included in a given bounded set. We will first of all consider finite sums of plane waves, and give a criterion on the amplitudes and directions of propagation of these plane waves which guarantees

Let $E$ be an elliptic curve defined over a number field $K$ with supersingular reduction at all primes of $K$ above $p$. If $K_\infty / K$ is a $\mathbb{Z}_p$-extension such that $E(K_\infty) [p^\infty]$ is finite and $H^2 (G_S (K_\infty), E [p^\infty]) = 0$, then we prove that the $\Lambda$-torsion subgroup of the Pontryagin dual of $\operatorname{Sel}_{p^\infty} (E / K_\infty)$ is pseudo-isomorphic

We define and study natural SU(2)-structures, in the sense of Conti–Salamon, on the total space $\mathcal{S}$ of the tangent sphere bundle of any given oriented Riemannian $3$-manifold $M$. We recur to a fundamental exterior differential system of Riemannian geometry. Essentially, two types of structures arise: the contact-hypo and the non-contact and, for each, we study the conditions for being hypo

The Hodge star mean curvature flow on a 3‑dimensional Riemannian or pseudo-Riemannian manifold is one of nonlinear dispersive curve flows in geometric analysis. Such a curve flow is integrable as its local differential invariants of a solution to the curve flow evolve according to a soliton equation. In this paper, we show that these flows on a 3‑sphere and 3‑dimensional hyperbolic space are integrable

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that $G = SU(2)$ acts on $M$ in a Hamiltonian manner, with nowhere vanishing moment map, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space, and the isotypical components are all finite dimensional. We consider the local

In this paper, we obtain a class of Virasoro modules by taking tensor products of the irreducible Virasoro modules $\Omega (\lambda, \alpha, h)$ defined in [CG1], with irreducible highest weight modules $V (\theta, h)$ or with irreducible Virasoro modules $\operatorname{Ind}_\theta (N)$ defined in [MZ2]. We obtain the necessary and sufficient conditions for such tensor product modules to be irreducible

We define a class of singularity on arbitrary pairs of a normal variety and an effective $\mathbb{R}$‑divisor on it, which we call pseudo‑$\operatorname{lc}$ in this paper. This is a generalization of the usual $\operatorname{lc}$ singularity of pairs and log canonical singularity of normal varieties introduced by de Fernex and Hacon. By giving examples of pseudo‑$\operatorname{lc}$ pairs which are

We show how to compute the Lagrangian Floer homology in the one-point blow up of the proper transform of Lagrangians submanifolds, solely in terms of information of the base manifold. As an example we present an alternative computation of the Lagrangian quantum homology in the one-point blow up of $(\mathbb{C}P^2, \omega_{FS})$ of the proper transform of the Clifford torus.

Let $\mathbb{K}$ be an imaginary quadratic field, $p$ a rational prime which splits in $\mathbb{K}$ into two distinct primes $\mathfrak{p}$ and $\mathfrak{\overline{p}}$, and $\mathbb{K}_\infty$ the unique $\mathbb{Z}_p$-extension of $\mathbb{K}$ unramified outside of $\mathfrak{p}$. For a finite abelian extension $\mathbb{L}$ of $\mathbb{K}$, we define $\mathbb{L}_\infty = \mathbb{LK}_\infty$, and

For $X$ a $d$-dimensional smooth projective variety over a field $k$ of characteristic $0$, using higher algebraic $K$-theory, we study the following two questions asked by Mark Green and Phillip Griffiths in chapter 10 of [9] (page 186-190): (1) For each positive integer $p$ satisfying $1 \leq p \leq d$, can one define the tangent space $TZ^p (X)$ to the cycle group $Z^p (X)$? (2) Obstruction issues

We study the Laplacian coflow and the modified Laplacian coflow of $G_2$-structures on the $7$-dimensional Heisenberg group. For the Laplacian coflow we show that the solution is always ancient, that is it is defined in some interval $(-\infty, T)$, with $0 \lt T \lt +\infty$. However, for the modified Laplacian coflow, we prove that in some cases the solution is defined only on a finite interval while

We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan.

Two new classes of summing multilinear operators, factorable $(q, p)$-summing operators and $(r; p, q)$-summing operators are studied. These classes are described in terms of factorization. It is shown that operators in the first (resp., the second) class admit the factorization through the injective tensor product of Banach spaces (resp., through some Banach lattices). Applications in different contexts

The purpose of this article is to investigate the properties of the category of mixed plectic Hodge structures defined by Nekovář and Scholl [NS1]. We give an equivalent description of mixed plectic Hodge structures in terms of the weight and partial Hodge filtrations. We also construct an explicit complex calculating the extension groups in this category.

It was proved by Huisken that a mean curvature flow converges to a self-shrinker in the Euclidean space after scaling when it develops a singularity of type I. In this paper, we study a coupled flow with a mean curvature flow and a Ricci flow, and generalize his result for this Ricci-mean curvature flow. Then, as a parabolic rescaling limit, we get a self-shrinker in a gradient shrinking Ricci soliton

We show how Conway’s multivariable potential function can be constructed using braids and the reduced Gassner representation. The resulting formula is a multivariable generalization of a construction, due to Kassel–Turaev, of the Alexander–Conway polynomial in terms of the Burau representation. Apart from providing an efficient method of computing the potential function, our result also removes the

We characterize unimodular solvable Lie algebras with Vaisman structures in terms of Kähler flat Lie algebras equipped with a suitable derivation. Using this characterization we obtain algebraic restrictions for the existence of Vaisman structures and we establish some relations with other geometric notions, such as Sasakian, co‑Kähler and left-symmetric algebra structures. Applying these results we

Let $M$ be a nilmanifold with a fundamental group which is free $2$-step nilpotent on at least $4$ generators. We will show that for any nonnegative integer n there exists a self-diffeomorphism $h_n$ of $M$ such that hn has exactly $n$ fixed points and any self-map $f$ of $M$ which is homotopic to $h_n$ has at least $n$ fixed points. We will also shed some light on the situation for less generators

We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by their mean curvature. We prove a differential Harnack inequality for any weakly convex solution to the mean curvature flow. As an application, by applying an Aleksandrov reflection argument, we classify convex, ancient solutions of the mean curvature flow on the sphere.

Let $(M, g)$ be an $n$-dimensional, compact Riemannian manifold. We will show that functions that are orthogonal to the first few Laplacian eigenfunctions have to have a large zero set. Let us assume $f \in C^0 (M)$ is orthogonal $\langle f, \phi_k \rangle = 0$ to all eigenfunctions $\phi_k$ with eigenvalue $\lambda_k \leq \lambda$. If $\lambda$ is large, then the function $f$ has to vanish on a large

We provide a partial classification of semistable Higgs bundles over simply connected Calabi–Yau manifolds. Applications to a conjecture about a special class of semistable Higgs bundles are given. In particular, the conjecture is proved for K3 and Enriques surfaces, and some related classes of surfaces.

We prove a conjecture of Gromov about non-free isometric immersions.

In this paper, using a spinorial approach, we generalize a theorem à la Alexandrov of Wang, Wang and Zhang [WWZ] to closed codimension-two spacelike submanifolds in the Minkowski spacetime for an adapted CMC condition.

Mather proved that the smooth stability of smooth maps between manifolds is a generic condition if and only if the pair of dimensions of the manifolds are ‘nice dimensions’ while topological stability is a generic condition in any pair of dimensions. And, by a result of du Plessis and Wall $C^1$-stability is also a generic condition precisely in the nice dimensions. We address the question of bi‑Lipschitz

We consider a supercritical branching process $(Z_n)$ in an independent and identically distributed random environment $\xi = (\xi_n)$. Let $W$ be the limit of the natural martingale $W_n = Z_n / E_\xi Z_n , n \geq 0$, where $E_\xi$ denotes the conditional expectation given the environment $\xi$. We find a necessary and sufficient condition for the existence of quenched weighted moments of $W$ of the

In [3], X. Chen proposed a continuity path aiming to attack the existence problem of the constant scalar curvature Kähler(cscK) metric. He also proved the openness of the path at $t \in (0, 1)$ by the standard implicit function theorem on solutions of fourth order PDE. However, the openness at $t = 0$ is quite different in nature and it is in fact a deformation result from the solution of a second

In [1], Guido De Philippis and Francesco Maggi proved global quadratic stability inequalities and derived explicit lower bounds for the first eigenvalues of the stability operators for all area-minimizing Lawson cones $M_{kh}$, except for those with\[(k, h), (h, k) \in S = \lbrace (3, 5), (2, 7), (2, 8), (2, 9), (2, 10), (2, 11) \rbrace \; \textrm{.}\]We proved the corresponding inequalities and lower

Given a submanifold $S \subset \mathbb{R}^n$ of codimension at least three, we construct an asymptotically Euclidean Riemannian metric on $\mathbb{R}^n$ with nonnegative scalar curvature for which the outermost apparent horizon is diffeomorphic to the unit normal bundle of $S$.

We study the asymptotic behavior of quantized Ding functionals along Bergman geodesic rays and prove that the slope at infinity can be expressed in terms of Donaldson–Futaki invariants and Chow weights. Based on the slope formula, we introduce a new algebro-geometric stability on Fano manifolds and show that the existence of anti-canonically balanced metrics implies our stability. The relation between