arXiv research

New CR almost Schur Lemma estimates curvature on compact manifolds.

In this paper, we study a general almost Schur Lemma on pseudo-Hermitian (2n+1)-manifolds $(M,J,θ)$ for $n\geq2$. When the equality of almost Schur inequality holds, we derive the contact form $θ$ is pseudo-Einstein and the pseudo-Hermitian scalar curvature is constant.

The paper proves new curvature estimates in quaternionic contact geometry.

In this paper, we prove almost Schur Lemma on closed smooth metric measure spaces, which implies the results of X. Cheng and De Lellis-Topping whenever the weighted function f is constant.

Researchers prove a stability result for a 3-sphere inequality, extending previous work.

In our previous paper in \cite{C}, we generalized the almost-Schur lemma of De Lellis and Topping for closed manifolds with nonnegative Rcci curvature to any closed manifolds. In this paper, we generalize the above results to symmetric $(2,0)$-tensors and give the applications including $r$th mean curvatures of closed …

In this short note we establish an integral geometric inequality in a smooth metric measure space of the nonnegative Bakry-Émery Ricci curvature. This result can be regarded as a mild generalization of the almost Schur theorem due to De Lellis and Topping (Calc. Var., DOI: 10.1007/s00526-011-0413-z).

Schur's lemma states that every Einstein manifold of dimension $n\geq 3$ has constant scalar curvature. Here $(M,g)$ is defined to be Einstein if its traceless Ricci tensor $$\Rico:=\Ric-\frac{R}{n}g$$ is identically zero. In this short note we ask to what extent the scalar curvature is constant if the traceless Ricci …

Proves estimate similar to De Lellis-Müller on Minkowski lightcone.

Paper extends Schur's theorem to spherical curves via monotonicity.

Deep neural networks are proven universally powerful using Koopman operator.

Paper derives a Reilly type integral formula and applies it to inequalities and eigenvalue problems.

After shortly recalling the construction of the Khovanov-Kuperberg algebras, we give a characterisation of indecomposable web-modules. It says that a web-module is indecomposable if and only if one can deduce it directly from the Kuperberg bracket (via a Schur lemma argument). The proofs relies on the construction of i…

The Schur's theorem of antiholomorphic type is proved for arbitrary almost Hermitian manifolds, namely: If a connected almost Hermitian manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then this curvature is a global constant.

Quantifies Schur's theorem for curves in CAT(k) spaces.

In this article, we define a symmetric 2-tensor canonically associated to Q-curvature called J-tensor on any Riemannian manifold with dimension at least three. The relation between J-tensor and Q-curvature is precisely like Ricci tensor and scalar curvature. Thus it can be interpreted as a higher-order analogue of Ricc…

An $η$-Einstein condition is introduced in the context of indefinite g.f.f-manifolds, and a few Schur-type lemmas for indefinite S-manifolds are provided.

Our main goal in this work is to deal with results concern to the $σ_2$-curvature. First we find a symmetric 2-tensor canonically associated to the $σ_2$-curvature and we present an Almost Schur Type Lemma. Using this tensor we introduce the notion of $σ_2$-singular space and under a certain hypothesis we prove a rigid…

Characterizes submanifolds with minimum ratio of diameter to focal radius.

In this article, we first establish the main tool - an integral formula for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly's original formula from \cite{Re2} and the recent result from \cite{QX}. It provides a robust tool for sub-static manifolds regardless…

Schwarz lemma extended to equality cases and curvature on manifolds.

We prove a version of Yau's Schwarz Lemma for general almost-complex manifolds equipped with Hermitian metrics. This requires an extension to this setting of the Laplacian comparison theorem. As an application we show that the product of two almost-complex manifolds does not admit any complete Hermitian metric with bis…

Paper studies curvature in Finsler geometry, proving curvature constancy under isotropy.

We study cohomologies on an almost complex manifold $(M, J)$, defined using the Nijenhuis-Lie derivations $\mathcal{L}_J$ and $\mathcal{L}_N$ induced from the almost complex structure $J$ and its Nijenhuis tensor $N$, regarded as vector-valued forms on $M$. We show how one of these, the $N$-cohomology $H^{\bullet}_N (M…

Using the method of De Lellis-Topping, we prove some almost Schur type results. For example, one of our results gives a quantitative measure of how close the higher mean curvature of a submanifold is to its average value. We also derive another sharp Andrews-De Lellis-Topping type inequality involving the Riemannian cu…

The $\bar{\partial}_{_{J}}$ operator over an almost complex manifold induces canonical connections of type $(0,1)$ over the bundles of $(p,0)$-forms. If the almost complex structure is integrable then the previous connections induce the canonical holomorphic structures of the bundles of $(p,0)$-forms. For $p=1$ we can …

Paper proves a new volume comparison theorem for Riemannian manifolds.

The paper solves a conjecture on almost complex 4-manifolds using refined Dolbeault cohomology.

The paper defines new types of positivity and proves properties of Schur forms for vector bundles.

One (actually, almost the only effective) way to prove formality of a differentiable manifold is to be able to produce a suitable derivation $δ$ such that $dδ$-lemma holds. We first show that such derivation $δ$ generates a (1,1)-tensor field (we denote it by $R$). Then, we show that the supercommutation of $d$ and $δ$…

Based on a self-contained, coordinate-free exposition of the necessary concepts and tools of spray and Finsler geometry (with detailed proofs), we derive new results among others on the consequences of the direction-independence of the Landsberg tensor and the stretch tensor of a Finsler manifold. We show that an at le…

In the paper, we give a Schur-Toponogov theorem in Riemannian geometry, which not only generalizes Schur's and Toponogov's theorem but also indicates their relation. Inspired by its proof, we also supply a new proof of Toponogov's theorem (in the large) in Alexandrov geometry.

Let M be an almost Hermitian manifold of dimension greater or equal to 6. The following theorems are proved: Theorem 1. If M is of pointwise constant θ-holomorphic sectional curvature for a number θ in (0,π/2) then M is of constant sectional curvature or a Kähler manifold of constant holomorphic sectional curvature. Th…

In the Friedmann Model of the universe, cosmologists assume that spacelike slices of the universe are Riemannian manifolds of constant sectional curvature. This assumption is justified via Schur's Theorem by stating that the spacelike universe is locally isotropic. Here we define a Riemannian manifold as almost locally…

Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.

We show that the Schur multiplier of $Sp(2g,\mathbb Z/D\mathbb Z)$ is $\mathbb Z/2\mathbb Z$, when $D$ is divisible by 4.

New unoriented versions of Schur and Bogomolov multipliers for finite groups.

Let X=G/P be cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) We say a Schubert class [S] is Schur rigid if the only irreducible subvarieties Y of X with homology class [Y] = r [S], for an integer r, are Schubert varieties. Robles and The identifie…

Efficiently computes embeddings for large graphs using coarsening.

Schur theorem proven for weakly Landsberg Finsler metrics.

New basis and Schur-Weyl duality for loop Hecke algebra defined.

New method connects neural networks to diagrammatic algebra.

Study framizations of algebras using Schur--Weyl duality and tied braids.

Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular Khovanov homology carries a commuting action of the symmetric group S_n…

Study proves uniqueness of Yang-Mills field tangent cones in arbitrary dimensions.

A new method for efficient portfolio optimization using graph structures.

The usual Gromoll-Meyer's generalized Morse lemma near degenerate critical points on Hilbert spaces, so called splitting lemma, is stated for at least $C^2$-smooth functionals. In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower…

The paper studies connectivity of Schur-Horn map images in real Grassmannians.