Andres Larrain-Hubach
For loop groups (free and based), we compute the exact order of the curvature operator of the Levi-Civita connection depending on a Sobolev space parameter. This extends results of Freed and Maeda-Rosenberg-Torres.
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Given a constant $k>1$ and a real valued function $K$ on the hyperbolic plane $\mathbb H^2$, we study the problem of finding, for any $ε\approx 0$, a closed and embedded curve $u^ε$ in $\mathbb H^2$ having geodesic curvature $k+εK(u^ε)$ at each point.
A consequence of the Cheeger-Gromoll splitting theorem states that for any open manifold $(M,x)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at $x$ that represent elements of $π_1(M,x)$ are contained in a bounded ball, then $π_1(M,x)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of $π_1(M,x)$ escape from any bounded metric balls at a sublinear rate with respect to their lengths, then $π_1(M,x)$ is virtually abelian.
This paper concerns complete noncompact manifolds with nonnegative Ricci curvature. Roughly, we say that M has the loops to infinity property if given any noncontractible closed curve, C, and given any compact set, K, there exists a closed curve contained in M\K which is homotopic to C. The main theorems in this paper are the following. Theorem I: If M has positive Ricci curvature then it has the loops to infinity property. Theorem II: If M has nonnegative Ricci curvature then it either has the loops to infinity property or it is isometric to a flat normal bundle over a compact totally geodesi…
We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.
We classify the connected pseudo-Riemannian manifolds of signature $(p,q)$ with $q\ge5$ so that at each point of $M$ the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least one point of $M$.
We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent bundle of some smooth manifold.
We study manifolds with almost nonnegative curvature operator (ANCO) and provide first examples of closed simply connected ANCO mannifolds that do not admit nonnegative curvature operator.
We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.
Direct proofs are given of Juhl's formulae for GJMS operators and Q-curvatures starting from the original construction of GJMS.
Planar loops with prescribed curvature via Hardy's inequality
本文研究具预定曲率的平面闭曲线存在性，提出基于变分法的方法，利用Hardy型不等式及相关函数空间进行证明。
The Weitzenböck curvature operators are the curvature terms of order zero that appear in the well known classical Weitzenböck formula. In this paper, we use the formalism of double forms to prove a simple formula for this operators and to study their geometric properties.
A note on parallel mean curvature surfaces and Codazzi operators
本文提出用Codazzi算子刻画的曲面内蕴Klotz-Osserman型定理，并利用Simons公式研究乘积空间中具非正Gauss曲率的平行平均曲率曲面。
In hyperbolic 3-space $\mathbb{H}^3$ surfaces of constant mean curvature $H$ come in three types, corresponding to the cases $0 \leq H < 1$, $H = 1$, $H > 1$. Via the Lawson correspondence the latter two cases correspond to constant mean curvature surfaces in Euclidean 3-space $\mathbb{E}^3$ with H=0 and $H \neq 0$, respectively. These surface classes have been investigated intensively in the literature. For the case $0 \leq H < 1$ there is no Lawson correspondence in Euclidean space and there are relatively few publications. Examples have been difficult to construct. In this paper we present …
Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If $π$ is a spacelike 2 plane, let $R(π)$ be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(-) has constant rank 2 and show these are geometrically realizable by hypersurfaces in flat spaces. We also classify the Ivanov-Petrova algebraic curvature tensors of rank 2; these are the algebraic curvature tensors of constant rank 2 such that the complex Jordan normal form of R(-) is constant.
Cheeger and Gromov showed that F-structures are related to collapse with a double-sided curvature bound. We define fibered F-structures and extend some of the Cheeger-Gromov results to the setting of collapse with a lower bound on the curvature operator.
Recently, a set of tools has been developed with the purpose of the study of Quantum Gravity. Until now, there have been very few attempts to put these tools into a rigorous mathematical framework. This is the case, for example, of the so called path bundle of a manifold. It is well known that this topological principal bundle plays the role of a universal bundle for the reconstruction of principal bundles and their connections. The path bundle is canonically endowed with a parallel transport and associated with it important types of derivatives have been considered by several authors: the Man…
We glue two manifolds which have curvature operators at least k (in the sense of eigenvalues) along their common boundary. We show that if the sum of the second fundamental forms of the boundary is positive semidefinite, then the curvature operator of the resulting manifold is at least k up to an arbitrarily small error term. Similar results hold for Ricci, scalar, bi, isotropic and flag curvature, respectively.
We show that any $k$ Osserman Lorentzian algebraic curvature tensor has constant sectional curvature and give an elementary proof that any local 2 point homogeneous Lorentzian manifold has constant sectional curvature. We also show that a Szabó Lorentzian covariant derivative algebraic curvature tensor vanishes.
We prove two splitting theorems, one topological, the other metric, for open manifolds with nonnegative sectional curvature.
Scalar curvature and Dirac operator for some singular spaces
本文研究了具有奇异黎曼度量的流形上标量曲率和狄拉克算子的存在条件。
Geodesic loops on tetrahedra in spaces of constant sectional curvature
本文研究了常截面曲率空间中四面体的测地环路，证明了球面空间中特定角度条件下简单测地环路的不存在性与存在性，以及双曲空间中正四面体上基于互质数对的简单测地环路的存在性。
In this paper, we study the following conjecture of Hamilton: Any compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein. We first derive several identities. Then we show that the conjecture is true under an additional condition. Furthermore, such a soliton must be of constant curvature.
The main purpose of this paper is to determine the admissible forms of the sectional curvature operator on a three-dimensional locally homogeneous Lorentzian manifolds.
This article aims to investigate the curvature operator of the second kind on Kähler manifolds. The first result states that an $m$-dimensional Kähler manifold with $\frac{3}{2}(m^2-1)$-nonnegative (respectively, $\frac{3}{2}(m^2-1)$-nonpositive) curvature operator of the second kind must have constant nonnegative (respectively, nonpositive) holomorphic sectional curvature. The second result asserts that a closed $m$-dimensional Kähler manifold with $\left(\frac{3m^3-m+2}{2m}\right)$-positive curvature operator of the second kind has positive orthogonal bisectional curvature, thus being biholo…
Using Weitzenböck techniques on any compact Riemannian spin manifold we derive inequalities that involve a real parameter and join the eigenvalues of the Dirac operator with curvature terms. The discussion of these inequalities yields vanishing theorems for the kernel of the Dirac operator $D$ and lower bounds for the spectrum of $D^2$ if the curvature satisfies certain conditions.
We show that a closed four-manifold with $4\frac{1}{2}$-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both $\mathbb{CP}^2$ and $\mathbb{S}^3 \times \mathbb{S}^1$ have $4\frac{1}{2}$-nonnegative curvature operator of the second kind. In higher dimensions $n\geq 5$, we show that closed Riemannian manifolds with $4\frac{1}{2}$-positive curvature operator of the second kind are homeomorphic to spherical space forms. These results are proved by showing that $4\frac{1}{2}$-positive curvature operator of the second kind…
We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A.
We study the positivity properties of the curvature operator for holomorphic Hermitian vector bundles. We obtain new characterization of semi-positive curvature operators for $(n,q)$ and $(p,n)$-forms by L2-estimates. The characterization of Nakano semi-positivity by $L^2$-estimate is already known. Applying our results, we give new characterizations of Nakano semi-negativity.
The curvature operator of the second kind in dimension three
本文研究三维Ricci流下第二类曲率算子的行为，通过用第一类曲率算子特征值显式表达第二类曲率算子特征值，并证明对所有$\a \in [1,5]$，$\a$-正/$\a$-非负的第二类曲率算子被Ricci流保持。
For $n\geq 7$, we give the optimal estimate for the second eigenvalue of Paneitz operators for compact $n$-dimensional submanifolds in an $(n+p)$-dimensional space form.
Let (M,g) be a complete noncompact riemannian manifold with bounded geometry and parallel Ricci curvature. We show that some operators, "affine" relatively to the Ricci curvature, are locally invertible, in some classical Sobolev spaces, near the metric g.
We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator $- 4 d^2/ds^2 + κ^2 (s)$ with potential given by the curvature of a closed curve.
Manifolds with harmonic Weyl curvature and curvature operator of the second kind
该研究证明了高维流形在特定曲率条件下的分类，并推广了相关领域的已有成果。
Manifolds with harmonic curvature and curvature operator of the second kind
该研究证明了具有特定调和曲率和第二类曲率算子条件的流形必为爱因斯坦流形，并进一步推广了Dai-Fu的结果。
Euler Characteristic of Closed Manifolds with Almost Nonnegative Curvature Operator
研究具有近非负曲率算子的闭合流形，解决Herrmann-Sebastian-Tuschmann关于其欧拉特征符号的问题，证明在特定条件下欧拉特征非负，并在无限基本群条件下证明欧拉特征、签名及$\widehat{A}$-特征 vanish。
We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric torsion free connection.
We study compact and simply-connected Riemannian manifolds with positive sectional curvature $K\ge 1.$ For a non-trivial homology class of lowest dimension in the space of loops based at a point $p$ or in the free loop space one can define a critical length ${\sf crl}_p\left(M,g\right)$ resp. ${\sf crl}\left(M,g\right).$ Then ${\sf crl}_p\left(M,g\right)$ equals the length of a geodesic loop and ${\sf crl}\left(M,g\right)$ equals the length of a closed geodesic. This is the idea of the proof of the existence of a closed geodesic of positive length presented by Birkhoff in case of a sphere and …
We associate to any Riemannian symmetric space (of finite or infinite dimension) a L$^*$-algebra, under the assumption that the curvature operator has a fixed sign. L$^*$-algebras are Lie algebras with a pleasant Hilbert space structure. The L$^*$-algebra that we construct is a complete local isomorphism invariant and allows us to classify Riemannian symmetric spaces with fixed-sign curvature operator. The case of nonpositive curvature is emphasized.
Decomposition of the Curvature Operator and Applications to the Hopf Conjecture
本文通过将曲率算子分解为 Hermitian 分量，建立截面曲率特征值判据与 Betti 数零化定理，证实小 Weyl 曲率情形的 Hopf 猜想并获得新刚性结果。
We give a new and complete proof of Hamilton's injectivity radius estimate for sequences with bounded and almost nonnegative curvature operators, unbounded diameters, and bump-like origins. Such sequences arise in particular from dilations about a singularity of the Ricci flow on a 3-manifold.
A Riemannian manifold is called IP, if the eigenvalues of its skew-symmetric curvature operator are pointwise constant. It was previously shown that for all n\ge 4, except n=7, any IP manifold either has constant curvature, or is a warped product, with some specific function, of a line and a space of constant curvature. We extend this result to the case n = 7, and also study 3-dimensional IP manifolds.
It is well known that the Jacobi operators completely determine the curvature tensor. The question of existence of a curvature tensor for given Jacobi operators naturally arises, which is considered and solved in the previous work. Unfortunately, although the published theorem is correct, its proof is incomplete because it contains some omissions, and the aim of this paper is to present a complete and accurate proof. We also generalize the main theorem to the case of indefinite scalar product space. Accordingly, we generalize the proportionality principle for Osserman algebraic curvature tenso…
We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing $L^{2}-$form. Moreover, we prove vanishing theorems for closed and co-closed conformal Killing $L^{2}-$forms on some complete Riemannian manifolds.
Poincaré Polynomials and Curvature Operators of Symmetric Spaces
该研究计算了所有紧不可约对称空间的曲率算子和庞加莱多项式的显式公式，利用量子数推导庞加莱多项式，并证明曲率算子最大特征值以爱因斯坦常数为上界且仅埃尔米特对称空间取等。
In this paper, we study the first eigenvalue of Jacobi operator on an $n$-dimensional non-totally umbilical compact hypersurface with constant mean curvature $H$ in the unit sphere $S^{n+1}(1)$. We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature $H$ and the dimension $n$.
The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersufaces in a locally flat projective manifold. We construct the GJMS operators and the Q-curvature for conformal Codazzi structures by using the ambient metric. We relate the total Q-curvature to the logarithmic coefficient in the volume expansion of the Blaschke metric, and derive the first and second variation formulas for a deformation of strictly convex domains.
We prove new lower bounds for the first eigenvalue of the Dirac operator on compact manifolds whose Weyl tensor or curvature tensor, respectively, is divergence free. In the special case of Einstein manifolds, we obtain estimates depending on the Weyl tensor.
In [AMW], it is proved that if a compact $3$-manifold has positive Ricci curvature and strictly convex boundary, then this manifold is diffeomorphic to the standard $3$-dimensional Euclidean disk. In this paper, we prove its higher-dimensional generalization: if a compact $n$-manifold has positive curvature operator and strictly convex boundary, then this manifold is diffeomorphic to the standard $n$-dimensional Euclidean disk.
In this short note, as a simple application of the strong result proved recently by Böhm and Wilking, we give a classification on closed manifolds with 2-nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifold (with dimension $\ge 3$) whose curvature operator is bounded and satisfies the pinching condition $R\ge δR_{I}>0$, for some $δ>0$, must be compact. This provides an intrinsic analogue of a result of Hamilton on convex hypersurfaces.
The purpose of this article is to initiate the investigation of the curvature operator of the second kind on Kähler manifolds. The main result asserts that a closed Kähler surface with six-positive curvature operator of the second kind is biholomorphic to $\mathbb{CP}^2$. It is also shown that a closed non-flat Kähler surface with six-nonnegative curvature operator of the second kind is either biholomorphic to $\mathbb{CP}^2$ or isometric to $\mathbb{S}^2 \times \mathbb{S}^2$.
We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators. We establish that on any manifold of dimension $n\geq 3$, there exist many metrics for which our invariants are nontrivial. We discuss new applications to curvature prescription problems.
A characterization of complex projective space via the Calabi curvature operator
该研究针对Kähler-Einstein流形，利用含Calabi曲率算子的新Bochner公式证明了Tachibana型结果，分离出复射影空间，解决了AIM研讨会提出的相关问题。
Let $M^n$ be a closed Riemannian manifold of dimension $n\geq 2$, with Ricci curvature $Ric \geq n-1$. We will show that any sphere of dimension $m$ in the space of closed loops on $M^n$ is homotopic to the sphere in the space of closed loops of length at most $8 πm$. It follows that the length of a shortest periodic geodesic on $M^n$ is bounded from above by $8 π(n-1)$.
Einstein manifolds and curvature operator of the second kind
本文利用 Bochner 技巧证明具有非负或一定阶数非负第二类曲率算子的紧致爱因斯坦流形必为常曲率空间，并结合已有结果给出统一判定条件。
We adapt the Bartnik method to provide a Hilbert manifold structure for the space of solutions, without KID's, to the vacuum constraint equations on compact manifold of any dimension $\geq 3$. In the course, we prove that some fibers of the scalar curvature or the constraint operator are Hilbert submanifolds. We also study some operators and inequalities related to the KID's operator. Finally we comment the adaptation to some non compact manifolds.
In this paper, we establish the non-positivity of the second eigenvalue of the Schrödinger operator $-\textrm{div}\big( P_r \nabla\cdot\big) - W_r^2$ on a closed hypersurface $Σ^n$ of $\mathbb{R}^{n+1}$, where $W_r$ is a power of the $(r+1)$-th mean curvature of $Σ^n$. In the case that this eigenvalue is null we have a characterization of the sphere. This generalizes a result of Evans and Loss proved for the Laplace-Beltrame operator penalized by the square of the mean curvature.
In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a simply-connected $n$-dimensional manifold of positive Ricci curvature $\textrm{Ric} \ge n-1$ has length $\le n π.$
In this paper we consider Riemannian manifolds $(M^n,g)$ of dimension $n \geq 5$, with semi-positive $Q$-curvature and non-negative scalar curvature. Under these assumptions we prove $(i)$ the Paneitz operator satisfies a strong maximum principle; $(ii)$ the Paneitz operator is a positive operator; and $(iii)$ its Green's function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive $Q$-curvature. Modifying the test function construction of Esposito-Robert, we show that it is possible to choose an initial conformal metric so that the…
Two Kinds of Curvature Operators in Dimension Four
本文证明了四维空间中两类曲率算子的不变恒等式，将第二类曲率算子与双正交截面曲率关联，并对满足 $K^\perp_{\max}\leq S/6$ 的闭定向四维流形完成分类，涵盖标量曲率消失的情况。
We investigate the curvature operator of the second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed Riemannian manifold with three-positive curvature operator of the second kind is diffeomorphic to a spherical space form, improving a recent result of Cao-Gursky-Tran assuming two-positivity. The second one states that a closed Riemannian manifold with three-nonnegative curvature operator of the second kind is either diffeomorphic to a spherical space form, or flat, or isometric to a quotient of a compact irreducible symmetric space. Thi…
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