The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
Feng-yu Wang - One of the best experts on this subject based on the ideXlab platform.
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Pointwise characterizations of curvature and Second Fundamental Form on Riemannian manifolds
Science China Mathematics, 2018Co-Authors: Feng-yu WangAbstract:Let M be a complete Riemannian manifold possibly with a boundary ∂M. For any C1-vector field Z, by using gradient/functional inequalities of the (reflecting) diffusion process generated by L:= Δ+Z, pointwise characterizations are presented for the Bakry-Emery curvature of L and the Second Fundamental Form of ∂M if it exists. These characterizations extend and strengthen the recent results derived by Naber for the uniForm norm ∥RicZ∥∞ on manifolds without boundaries. A key point of the present study is to apply the asymptotic Formulas for these two tensors found by the first author, such that the proofs are significantly simplified.
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for the Second Fundamental Form
2010Co-Authors: Feng-yu WangAbstract:Let M be a compact Riemannian manifold with bound- ary @M and L = �+Z for a C 1 -vector field Z on M. Several equivalent statements, including the gradient and Poincare/log-Sobolev type in- equalities of the Neumann semigroup generated by L, are presented for lower bound conditions on the curvature of L and the Second Fundamental Form of @M. The main result not only generalizes the corresponding known ones on manifolds without boundary, but also clarifies the role of the Second Fundamental Form in the analysis of the Neumann semigroup. Moreover, the Levy-Gromov isoperimetric inequality is also studied on manifolds with boundary.
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Semigroup properties for the Second Fundamental Form
Documenta Mathematica, 2010Co-Authors: Feng-yu WangAbstract:Let M be a compact Riemannian manifold with boundary ∂M and L = δ+Z for a C-vector field Z onM . Several equivalent statements, including the gradient and Poincare/log-Sobolev type inequalities of the Neumann semigroup generated by L, are presented for lower bound conditions on the curvature of L and the Second Fundamental Form of ∂M . The main result not only generalizes the corresponding known ones on manifolds without boundary, but also clarifies the role of the Second Fundamental Form in the analysis of the Neumann semigroup. Moreover, the Levy-Gromov isoperimetric inequality is also studied on manifolds with boundary. 2010 Mathematics Subject Classification: 60J60, 58G32.
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Semigroup Properties for the Second Fundamental Form
arXiv: Probability, 2009Co-Authors: Feng-yu WangAbstract:Let $M$ be a compact Riemannian manifold with boundary $\pp M$ and $L= \DD+Z$ for a $C^1$-vector field $Z$ on $M$. Several equivalent statements, including the gradient and Poincar\'e/log-Sobolev type inequalities of the Neumann semigroup generated by $L$, are presented for lower bound conditions on the curvature of $L$ and the Second Fundamental Form of $\pp M$. The main result not only generalizes the corresponding known ones on manifolds without boundary, but also clarifies the role of the Second Fundamental Form in the analysis of the Neumann semigroup. Moreover, the L\'evy-Gromov isoperimetric inequality is also studied on manifolds with boundary.
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heat kernel inequalities for curvature and Second Fundamental Form
2009Co-Authors: Feng-yu WangAbstract:On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup is proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for $p_t(x,y)$ the Neumann heat kernel w.r.t. a volume type measure $\mu$ and for $K$ a constant, the curvature condition $\Ric-\nn Z\ge K$ together with the convexity of the boundary is equivalent to the heat kernel entropy inequality $$\int_M p_t(x,z)\log \ff{p_t(x,z)}{p_t(y,z)} \mu(\d z)\le \ff{K\rr(x,y)^2}{2(\e^{2Kt}-1)}, t>0, x,y\in M,$$ where $\rr$ is the Riemannian distance. The main result is partly extended to manifolds with non-convex boundary and applied to derive the HWI inequality.
Shujie Zhai - One of the best experts on this subject based on the ideXlab platform.
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Submanifolds with Parallel Möbius Second Fundamental Form in the Unit Sphere
Results in Mathematics, 2018Co-Authors: Shujie ZhaiAbstract:In this paper, we establish a complete classification of umbilic-free submanifolds of the unit sphere with parallel Mobius Second Fundamental Form and arbitrary codimension.
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On submanifolds with parallel Möbius Second Fundamental Form in the unit sphere
International Journal of Mathematics, 2014Co-Authors: Shujie Zhai, Changping WangAbstract:In this paper, we study umbilic-free submanifolds of the unit sphere with parallel Mobius Second Fundamental Form. As one of our main results, we establish a complete classification for such submanifolds under the additional condition of codimension two.
Felix Lubbe - One of the best experts on this subject based on the ideXlab platform.
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A Bernstein Theorem for Minimal Maps with Small Second Fundamental Form
Results in Mathematics, 2018Co-Authors: Felix LubbeAbstract:We consider minimal maps $f:M\to N$ between Riemannian manifolds $(M,\mathrm{g}_M)$ and $(N,\mathrm{g}_N)$, where $M$ is compact and where the sectional curvatures satisfy $\sec_N\le \sigma\le \sec_M$ for some $\sigma>0$. Under certain assumptions on the differential of the map and the Second Fundamental Form of the graph $\Gamma(f)$ of $f$, we show that $f$ is either the constant map or a totally geodesic isometric immersion.
Xiaoxiang Jiao - One of the best experts on this subject based on the ideXlab platform.
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classification of conFormal minimal immersions from s 2 to g 2 n mathbb c with parallel Second Fundamental Form
arXiv: Differential Geometry, 2019Co-Authors: Xiaoxiang JiaoAbstract:In this paper, we determine all conFormal minimal immersions of 2-spheres in complex Grassmann manifold $G(2,N; \mathbb{C})$ with parallel Second Fundamental Form.
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Classification of conFormal minimal immersions from $S^2$ to $G(2,N;\mathbb{C})$ with parallel Second Fundamental Form
arXiv: Differential Geometry, 2019Co-Authors: Xiaoxiang JiaoAbstract:In this paper, we determine all conFormal minimal immersions of 2-spheres in complex Grassmann manifold $G(2,N; \mathbb{C})$ with parallel Second Fundamental Form.
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Minimal surfaces in symmetric spaces with parallel Second Fundamental Form
Proceedings - Mathematical Sciences, 2017Co-Authors: Xiaoxiang JiaoAbstract:In this paper, we study geometry of isometric minimal immersions of Riemannian surfaces in a symmetric space by moving frames and prove that the Gaussian curvature must be constant if the immersion is of parallel Second Fundamental Form. In particular, when the surface is \(S^2\), we discuss the special case and obtain a necessary and sufficient condition such that its Second Fundamental Form is parallel. We also consider isometric minimal two-spheres immersed in complex two-dimensional Kahler symmetric spaces with parallel Second Fundamental Form, and prove that the immersion is totally geodesic with constant Kahler angle if it is neither holomorphic nor anti-holomorphic with Kahler angle \(\alpha \ne 0\) (resp. \(\alpha \ne \pi \)) everywhere on \(S^2\).
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Minimal two-spheres in G (2, 4; ( C )) with parallel Second Fundamental Form
Mathematical Notes, 2017Co-Authors: Wenjuan Zhang, Xiaoxiang JiaoAbstract:In this paper, we give a classification theorem of minimal two-spheres in G(2, 4; (C)) with parallel Second Fundamental Form. Moreover, we also consider some special holomorphic two-spheres in G(2, n; (C)) and give the corresponding conditions of the parallel Second Fundamental Form.
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On ConFormal Minimal Immersions of Two-Spheres in a Complex Hyperquadric with Parallel Second Fundamental Form
The Journal of Geometric Analysis, 2014Co-Authors: Xiaoxiang JiaoAbstract:In this paper we determine all conFormal minimal immersions of two-spheres in complex hyperquadric \(Q_n\) with parallel Second Fundamental Form.
Steven Verpoort - One of the best experts on this subject based on the ideXlab platform.
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The mean curvature of the Second Fundamental Form of a hypersurface
2010Co-Authors: Stefan Haesen, Steven VerpoortAbstract:An expression for the first variation of the area functional of the Second Fundamental Form is given for a hypersurface in a semiRiemannian space. The concept of the “mean curvature of the Second Fundamental Form” is then introduced for hypersurfaces in semiRiemannian spaces. Some characterisations of extrinsic hyperspheres in terms of this curvature are given. MSC 2000: 53B25 (primary), 53A10, 53C42 (Secondary)
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On the Area Functional of the Second Fundamental Form of Ovaloids
Mathematical Proceedings of the Royal Irish Academy, 2009Co-Authors: Steven VerpoortAbstract:The expression for the variation of the area functional of the Second Fundamental Form of a hypersurface in a Euclidean space involves the so-called 'mean curvature of the Second Fundamental Form.' Several new characteristic properties of (hyper) spheres in which the mean curvature of the Second Fundamental Form occurs are given. In particular, it is shown that the spheres are the only ovaloids that are a critical point of the area functional of the Second Fundamental Form under various constraints.
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on the inner curvature of the Second Fundamental Form of a surface in the hyperbolic space
arXiv: Differential Geometry, 2008Co-Authors: Steven VerpoortAbstract:The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite Second Fundamental Form. It is shown that several conditions on the Gaussian curvature of the Second Fundamental Form can be satisfied only by extrinsic spheres.
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THE MEAN CURVATURE OF THE Second Fundamental Form
Houston Journal of Mathematics, 2008Co-Authors: Stephan Haesen, Steven Verpoort, Leopold VerstraelenAbstract:The critical points of the area functional of the Second Fundamental Form of Riemannian surfaces in three-dimensional semi-Riemannian manifolds are determined. They are characterized by the vanishing of a scalar function, which will be called the mean curvature of the Second Fundamental Form. A property which involves this new mean curvature is distinctive for totally umbilical surfaces
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The Mean Curvature of the Second Fundamental Form of a hypersurface
arXiv: Differential Geometry, 2007Co-Authors: Stephan Haesen, Steven VerpoortAbstract:An expression for the first variation of the area functional of the Second Fundamental Form is given for a hypersurface in a semi-Riemannian space. The concept of the "mean curvature of the Second Fundamental Form" is then introduced. Some characterisations of extrinsic hyperspheres in terms of this curvature are given.
