scholarly journals Lyapunov spectrum of invariant subbundles of the Hodge bundle

2012 ◽  
Vol 34 (2) ◽  
pp. 353-408 ◽  
Author(s):  
GIOVANNI FORNI ◽  
CARLOS MATHEUS ◽  
ANTON ZORICH
Keyword(s):  
Lyapunov Exponents ◽  
Fundamental Form ◽  
Moduli Space ◽  
Second Fundamental Form ◽  
Lyapunov Spectrum ◽  
Partial Sums ◽  
Probability Measures ◽  
The Second Fundamental Form ◽  
Special Cases ◽  
Invariant Subbundles

AbstractWe study the Lyapunov spectrum of the Kontsevich–Zorich cocycle on SL(2,ℝ)-invariant subbundles of the Hodge bundle over the support of SL(2,ℝ)-invariant probability measures on the moduli space of Abelian differentials. In particular, we prove formulas for partial sums of Lyapunov exponents in terms of the second fundamental form (the Kodaira–Spencer map) of the Hodge bundle with respect to the Gauss–Manin connection and investigate the relations between the central Oseledets subbundle and the kernel of the second fundamental form. We illustrate our conclusions in two special cases.

scholarly journals Biwarped product submanifolds of a Kähler manifold

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2349-2365 ◽  
Author(s):  
Hakan Taştan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Kähler Manifold ◽  
Second Fundamental Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Equality Case ◽  
The Second Fundamental Form ◽  
Special Cases ◽  
Necessary And Sufficient

We study biwarped product submanifolds which are special cases of multiply warped product submanifolds in K?hler manifolds. We observe the non-existence of such submanifolds under some circumstances. We show that there exists a non-trivial biwarped product submanifold of a certain type by giving an illustrate example. We also give a necessary and sufficient condition for such submanifolds to be locally trivial. Moreover, we establish an inequality for the squared norm of the second fundamental form in terms of the warping functions for such submanifolds. The equality case is also discussed.

scholarly journals First eigenvalue of submanifolds in Euclidean space

2000 ◽  
Vol 24 (1) ◽  
pp. 43-48
Author(s):  
Kairen Cai
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
The Second Fundamental Form ◽  
Compact Submanifold

We give some estimates of the first eigenvalue of the Laplacian for compact and non-compact submanifold immersed in the Euclidean space by using the square length of the second fundamental form of the submanifold merely. Then some spherical theorems and a nonimmersibility theorem of Chern and Kuiper type can be obtained.

Complete 3-dimensional λ-translators in the Euclidean space ℝ4

2021 ◽  
pp. 1-54
Author(s):  
Zhi Li ◽  
Guoxin Wei ◽  
Gangyi Chen
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
3 Dimensional ◽  
The Second Fundamental Form

In this paper, we obtain the classification theorems for 3-dimensional complete [Formula: see text]-translators [Formula: see text] with constant squared norm [Formula: see text] of the second fundamental form and constant [Formula: see text] in the Euclidean space [Formula: see text].

scholarly journals A priori estimates for a relativistic liquid with free surface boundary

2019 ◽  
Vol 16 (03) ◽  
pp. 401-442
Author(s):  
Daniel Ginsberg
Keyword(s):  
Free Surface ◽  
Fundamental Form ◽  
A Priori Estimates ◽  
Fluid Velocity ◽  
A Priori ◽  
Second Fundamental Form ◽  
Energy Estimates ◽  
Surface Boundary ◽  
The Second Fundamental Form ◽  
Sobolev Norms

We prove energy estimates for a relativistic free liquid body with sufficiently small fluid velocity in a general Lorentz spacetime. These estimates control Sobolev norms of the fluid velocity and enthalpy in the interior as well as Sobolev norms of the second fundamental form on the boundary. These estimates are generalizations of the energy estimates of Christodoulou and Lindblad [D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Commun. Pure Appl. Math. 53(12) (2000) 1536–1602] and rely on elliptic estimates which only require bounds for the second fundamental form of the time slices of the free boundary.

scholarly journals A characterization of complete Riemannian manifolds minimally immersed in the unit sphere

1993 ◽  
Vol 131 ◽  
pp. 127-133 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Clifford Torus ◽  
Veronese Surface ◽  
The Second Fundamental Form ◽  
Complete Riemannian Manifolds

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totallygeodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

scholarly journals Space-like submanifolds with parallel mean curvature in de Sitter spaces

2000 ◽  
Vol 69 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Chongzhen Ouyang ◽  
Zhenqi Li
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Mean Curvature Vector ◽  
De Sitter ◽  
Parallel Mean Curvature Vector ◽  
Complete Space ◽  
The Second Fundamental Form ◽  
Parallel Mean Curvature

AbstractThis paper investigates complete space-like submainfold with parallel mean curvature vector in the de Sitter space. Some pinching theorems on square of the norm of the second fundamental form are given

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