Related papers: Rigidity and vanishing theorems for complete trans…
In this paper, we compute the first and second variation formulas for the F-functional of translating solitons and study the Hamiltonian L-stability of Lagrangian translating solitons. We prove that any Lagrangian translating soliton is…
General relativity is a covariant theory of two transverse, traceless graviton degrees of freedom. According to a theorem of Hojman, Kuchar, and Teitelboim, modifications of general relativity must either introduce new degrees of freedom or…
By carrying out refined point-wise estimates for the mean curvature, we prove better rigidity theorems of Lagrangian and symplectic translating solitons.
When $X=\Gamma\backslash \H^n$ is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of $L^2$ harmonic forms vanish. In this paper, we show rigidity…
We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity…
In this article, we first consider the $L^{2}$ \textit{Morse-Novikov cohomology} on a complete Riemannian manifold $M$ equipped with a parallel $1$-form which includes Vaisman manifold. Based on a vanishing theorem of $L^{2}$…
This note is concerned with some essential properties (optimal isoperimetry, first variation, and monotonicity formula) of the so-called $[0,1)\ni\gamma$-torsional rigidity $\mathcal{T}_{\gamma,\mathsf{g}}$ on a complete Riemannian…
For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality…
Solid-like behavior at low energies and long distances is usually associated with the spontaneous breaking of spatial translations at microscopic scales, as in the case of a lattice of atoms. We exhibit three quantum field theories that are…
We use bounded cohomology to define a notion of volume of an SO(n,1)-valued representation of a lattice SO(n,1) and, using this tool, we give a complete proof of the volume rigidity theorem of Francaviglia and Klaff in this setting. Our…
We study succinctness as a measure of the expressive power of transformers. Succinctness -- how compactly a formalism can describe a language relative to other formalisms -- is a classical notion in logic and automata theory. We prove that…
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…
We study supertranslations at spatial and future timelike infinity in the first-order formalism. We relax the Ashtekar-Engle-Sloan boundary conditions to allow supertranslations at the spatial infinity and obtain the precise form of the…
In this paper, we completely classify $3$-dimensional complete self-expanders with constant norm $S$ of the second fundamental form and constant $f_{3}$ in Euclidean space $\mathbb R^{4}$, where $h_{ij}$ are components of the second…
We study pseudo-Riemanniasn manifolds $(M,g)$ with transitive group of conformal transformation which is essential, i.e. does not preserves any metric conformal to $g$. All such manifolds of Lorentz signature with non exact isotropy…
In this paper, we show several vanishing type theorems for $p$-harmonic $\ell$-forms on Riemannian manifolds ($p\geq2$). First of all, we consider complete non-compact immersed submanifolds $M^n$ of ${N}^{n+m}$ with flat normal bundle, we…
Every homomorphism from finite index subgroups of a universal lattices to mapping class groups of orientable surfaces (possibly with punctures), or to outer automorphism groups of finitely generated nonabelian free groups must have finite…
We prove the Second Vanishing Theorem for local cohomology modules of an unramified regular local ring in its full generality and provide a new proof of the Second Vanishing Theorem in prime characteristic $p$. As an application of our…
Very narrow spatial bright solitons in (1+1)D and (2+1)D versions of cubic-quintic and full saturable models are studied, starting from the full system of the Maxwell's equations, rather than from the paraxial (NLS) approximation. For the…
We provide a characterization of discrete sets $\Lambda \subset \mathbb{R}$ that admit a function whose $\Lambda$-translates are complete in the Hardy space $H^1(\mathbb{R})$. In particular, we show that such a set cannot be uniformly…