Artem Pulemotov
The paper establishes a version of the Hopf boundary point lemma for sections of a vector bundle over a manifold with boundary. This result may be viewed as a counterpart to the tensor maximum principle obtained by R. Hamilton in 1986. Potential applications include the study of various geometric flows and the construction of invariant sets for geometric boundary value problems.
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The Hopf Lemma for second order elliptic operators is proved to hold in domains with $C^{1,α}$, and even less regular, boundaries. It need not hold for $C^1$ boundaries. Corresponding results are proved for second order parabolic operators.
A classical theorem of A.D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed.
In [1], Theorem 3, the authors proved, in one dimension, a generalization of the Hopf Lemma, and the question arose if it could be extended to higher dimensions. In this paper we present two conjectures as possible extensions, and give a very partial answer. We write this paper to call attention to the problem.
A classical result of A.D. Alexandrov states that a connected compact smooth $n-$dimensional manifold without boundary, embedded in $\Bbb R^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_{n+1}=$constant in case $M$ satisfies: for any two points $(X', X_{n+1})$, $(X', \hat X_{n+1})$ on $M$, with $X_{n+1}>\hat X_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the H…
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.
We demonstrate that it is conceptually and computationally favorable to regard spin-weighted spherical harmonics as vector valued functions on the total space $SO(3)$ of the Hopf bundle, satisfying a covariance condition with respect to the gauge group $U(1)$ of this bundle. A key role is played by the invariant connection form of the principle Hopf bundle, known to physicists from the geometry behind magnetic monopoles.
We establish Hörmander-type $L^2$-estimates for the $\overline{\partial}$-operators that hold uniformly for all nontrivial flat holomorphic line bundles on compact Kähler manifolds. Our result can be regarded as a $\overline{\partial}$-version of Ueda's lemma on the operator norm of Čech coboundaries for flat line bundles and indeed recovers the original version of Ueda's lemma for compact Kähler manifolds. A partial generalisation for $(p,0)$-forms on Ricci-flat manifolds is also given.
We show, using two different approaches, that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian submersion from the universal covering space of the unit tangent bundle onto the two-sphere. A hyperbolic counterpart dealing with the tangent bundle of a hyperbolic plane is also presented.
We describe a diagram containing the zero sets of the moment maps associated to the diagonal U(1) and Sp(1) actions on the quaternionic projective space HP^n. These sets are related both to focal sets of submanifolds and to Sasakian-Einstein structures on induced Hopf bundles. As an application, we construct a complex structure on the Stiefel manifolds V_2 (C^{n+1}) and V_4 (R^{n+1}), the one on the former manifold not being compatible with its known hypercomplex structure.
The sub-Finslerian geometry means that the metric $F$ is defined only on a given subbundle of the tangent bundle, called a horizontal bundle. In the paper, a version of the Hopf-Rinow theorem is proved in the case of sub-Finslerian manifolds, which relates the properties of completeness, geodesically completeness, and compactness. The sub-Finsler bundle, the exponential map and the Legendre transformation are deeply involved in this investigation.
A Mall bundle on a Hopf manifold H is a holomorphic vector bundle whose pullback to the universal cover of H is trivial. We define resonant and non-resonant Mall bundles, generalizing the notion of the resonance in ODE, and prove that a non-resonant Mall bundle always admits a flat holomorphic connection. We use this observation to prove a version of Poincare-Dulac linearization theorem, showing that any non-resonant invertible holomorphic contraction of a complex space is linear in appropriate holomorphic coordinates. We define the notion of resonance in Hopf manifolds, and show that all non-…
For the ``Hopf bundle'' $S^1\to S^{2n,1} \to {\mathbb H}^n$, horizontal lifts of simple closed curves are studied. Let $γ$ be a piecewise smooth, simple closed curve on a complete totally geodesic surface $S$ in the base space. Then the holonomy displacement along $γ$ is given by $$ V(γ)=e^{λA(γ) i} $$ where $A(γ)$ is the area of the region on the surface $S$ surrounded by $γ$; $λ=1/2 $ or 0 depending on whether $S$ is a complex submanifold or not. We also carry out a similar investigation for the complex Heisenberg group ${\mathbb R} \to {\mathcal H}^{2n+1} \to {\mathbb C}^n$.
Let $A$ be a diagonal linear operator on $\C^n$, with all eigenvalues satisfying $0<|α_i|<1$, and $M = (\C^n\backslash 0)/<A>$ the corresponding Hopf manifold. We show that any stable holomorphic bundle on $M$ can be lifted to a $G$-equivariant coherent sheaf on $\C^n$, where $G=(\C^*)^l$ is a Lie group acting on $\C^n$ and containing $A$. This is used to show that all stable bundles on $M$ are filtrable, that is, admit a filtration by a sequence $F_i$ of coherent sheaves, with all subquotients $F_i/F_{i-1}$ of rank 1.
The Hopf Algebraic Structure of Finitely Supported Currents on a Lie Group
研究有限支撑电流在李群上的霍普夫代数结构，通过卷积和楔积定义乘法与余乘法，揭示其作为有限支撑截面函子的应用及具体公式。
Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations. Given an oriented Euclidean bundle $V \to X$ of rank divisible by four over a finite complex $X$ we derive a stable decomposition result for vector bundles over the sphere bundle $\mathord{\mathbb S}( \mathbb{R} \oplus V)$ in terms of vector bundles and Clifford module bundles over $…
For two complex vector bundles admitting a homomorphism with isolated singularities between them, we establish a Poincaré-Hopf type formula for the difference of the Chern character numbers of these two vector bundles. As a consequence, we extend the original Poincaré-Hopf index formula to the case of complex vector fields (to appear in Mathematische Zeitschrift)
The Hopf theorem states that homotopy classes of continuous maps from a closed connected oriented smooth $n$-manifold $M$ to the $n$-sphere are classified by their degree. Such a map is equivalent to a section of the trivial $n$-sphere bundle over $M$. A generalization of the Hopf theorem is obtained for sections of nontrivial oriented $n$-sphere bundles over $M$.
This paper is a continuation of the paper F. A. Arias and M. Malakhaltsev "A generalization of the Gauss-Bonnet and Hopf-Poincaré theorems", ArXiv:1510.01395 [MathDG] 5 Oct 2015. Let $π: E \to M$ be a locally trivial fiber bundle over a two-dimensional manifold $M$, and $Σ\subset M$ be a discrete subset. A subset $Q \subset E$ is called an $n$-sheeted branched section of the bundle $π$ if $Q' = π^{-1}(M \setminus Σ) \cap Q$ is a $n$-sheeted covering of $M \setminus Σ$. The set $Σ$ is called the singularity set of the branched section $Q$. We define the index of a singularity point of a branche…
For two complex vector bundles admitting a homomorphism between them, a Poincaré-Hopf formula for the difference of the Chern character numbers of these two vector bundles with isolated singularities is established by Huitao Feng, Weiping Li and Weiping Zhang. This article extend their reslut about Poincaré-Hopf type formula for the difference of the Chern character numbers to the non-isolated singularities. A special connection $\widetilde{\nabla}_{1}^{E}$ be constructed, the Chern character ${\rm ch}(E,\widetilde{\nabla}_{1}^{E})$ is the key role for the non-isolated singularities. As a cons…
On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups
本文完全刻画了具有双曲基本群的闭可定向非球面流形上圆丛的绝对度大于一的自映射存在性，证明其等价于丛虚拟平凡，并由此验证了Hopf问题的强版本，同时构造了关于映射度单调的新数值不变量。
We consider a locally trivial fiber bundle $π: E \to M$ over a compact oriented two-dimensional manifold $M$, and a section $s$ of this bundle defined over $M \setminus Σ$, where $Σ$ is a discrete subset of $M$. We call the set $Σ$ the set of singularities of the section $s : M \setminus Σ\to E$. We assume that the behavior of the section $s$ at the singularities is controlled in the following way: $s(M \setminus Σ)$ coincides with the interior part of a surface $S \subset E$ with boundary $\partial S$, and $\partial S$ is $π^{-1}(Σ)$. For such sections $s$ we define an index of $s$ at a point…
We prove that the Hopf vector field is a unique one among geodesic covariantly normal unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation using standard approach from the theory of submanifolds and find exact boundaries for the sectional curvature of the Hopf vector field.
Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein type theorems in higher codimension for minimal maps between Riemannian manifolds.
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