Related papers: Vanishing theorems for associative submanifolds

In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional $C^\infty$-manifolds in convenient calculus. More precisely, we discuss the smoothing of maps,…

Let $(M,g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2\leq n\leq 6$. Our main theorem generalizes the solution of S.-T. Yau's conjecture on the abundance of minimal surfaces and builds on a result of M. Gromov. Suppose…

Junyan Cao has obtained a very general vanishing theorem, valid on any compact K\"ahler manifold, for the cohomology groups with values in a pseudoeffective line bundle twisted by the associated multiplier ideal sheaf. In this note, we give…

We show that for generic choices of parameters the moduli spaces of periodic monopoles (with singularities), i.e. monopoles on $\mathbb{R}^{2} \times \mathbb{S}^{1}$ possibly singular at a finite collection of points, are either empty or…

We investigate the local deformation space of 3-dimensional cone-manifold structures of constant curvature $\kappa \in \{-1,0,1\}$ and cone-angles $\leq \pi$. Under this assumption on the cone-angles the singular locus will be a trivalent…

We address the following conjecture about the existence of common zeros for commuting vector fields in dimension three: if $X,Y$ are two $C^1$ commuting vector fields on a $3$-manifold $M$, and $U$ is a relatively compact open such that $X$…

Let $M$ be a locally symmetric irreducible closed manifold of dimension $\ge 3$. A result of Borel [Bo] combined with Mostow rigidity imply that there exists a finite group $G = G(M)$ such that any finite subgroup of $\text{Homeo}^+(M)$ is…

An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold $M$ with holonomy group of shrinkable dimension (or discompacit\'e in…

The Bochner technique is a classical tool in global differential geometry for proving vanishing and rigidity results by exploiting curvature conditions. Building on recent extensions of this method to complete non-compact settings by…

In this paper, we revise some results on rigidity and vanishing properties obtained by \textit{Cuong et.al} in \cite{CDS24} on $n$-dimensional totally real minimal submanifolds $M$ immersed in complex space forms $\widetilde{M}^n(c)$, for…

We prove several vanishing theorems for the cohomology of balanced hyperbolic manifolds that we introduced in our previous work and for the $L^2$ harmonic spaces on the universal cover of these manifolds. Other results include a Hard…

Consider a finite-dimensional algebra $A$ and any of its moduli spaces $\mathcal{M}(A,\mathbf{d})^{ss}_{\theta}$ of representations. We prove a decomposition theorem which relates any irreducible component of…

Let $N$ be a Riemannian manifold and consider a stationary union of three or more $C^{1,\mu}$ hypersurfaces-with-boundary $M_k$ in $N$ with a common boundary $\Gamma$. We show that if $N$ is smooth, then $\Gamma$ is smooth and each $M_k$ is…

Given scheme-theoretic equations for a nonsingular subvariety, we prove that the higher cohomology groups for suitable twists of the corresponding ideal sheaf vanish. From this result, we obtain linear bounds on the multigraded…

This article constructs examples of associative submanifolds in $G_2$-manifolds obtained by resolving $G_2$-orbifolds using Joyce's generalised Kummer construction. As the $G_2$-manifolds approach the $G_2$-orbifolds, the volume of the…

We introduce a method to construct closed rigid associative submanifolds in twisted connected sum $G_2$-manifolds. More precisely, we prove a gluing theorem of asymptotically cylindrical (ACyl) associative submanifolds in ACyl…

We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) $G_2$-manifolds. In the AC case, we show that if the rate of convergence $\nu$ to the cone at infinity is generic in a precise sense and lies…

We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for $\underline{E}G$ and satisfy properties (M) and (NM). Among the examples of groups satisfying…

We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact…

Let $(R,\fr m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M, N$ two finitely generated $R$-modules. The first result of this paper is to prove a vanishing theorem for generalized local cohomology modules which says that…