We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n−1)-submanifold S . The main result is that the fundamental group of M \S is relatively hyperbolic, relative to fundamental groups of the ends of M \S , and M \S admits a complete finite volume A -regular Riemannian metric of negative sectional curvature. It fo...

The Teichmüller space Teich(S) of a surface S in genus g > 1 is a totally real submanifold of the quasifuchsian space QF(S). We show that the determinant of the Laplacian det′(∆) on Teich(S) has a unique holomorphic extension to QF(S). To realize this holomorphic extension as the determinant of differential operators on S, we introduce a holomorphic family {∆μ,ν} of elliptic second order differ...

We give asymptotically tight estimate of tangent space variation on a Riemannian submanifold of Euclidean space with respect to the local feature size of the submanifold. We show that the result is a consequence of structural properties of local feature size function of the Riemannian submanifold.

A Zoll metric is a Riemannian metric whose geodesics are all circles of equal length. Via the twistor correspondence of LeBrun and Mason, a Zoll metric on the sphere S corresponds to a family of holomorphic disks in CP2 with boundary in a totally real submanifold P ⊂ CP2. In this paper, we show that for a fixed P ⊂ CP2, such a family is unique if it exists, implying that the twistor corresponde...

An analytic pair of dimension n and center V is a pair (V,M) where M is a complex manifold of (complex) dimension n and V ⊂ M is a closed totally real analytic submanifold of dimension n. To an analytic pair (V,M) we associate the class U (V,M) of the functions u : M → [0,π/4[ which are plurisubharmonic in M and such that u(p)= 0 for each p ∈ V . If U (V,M) admits a maximal function u, the trip...

We introduce the process of symplectic reduction along a submanifold as uniform approach to taking quotients in geometry. This construction holds categories smooth manifolds, complex analytic spaces, and algebraic varieties, has an interpretation terms derived stacks shifted It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein pre-images Poisson transversals under moment maps...

Using the submanifold quantum mechanical scheme, the restricted Dirac operator in a submanifold is defined. Then it is shown that the zero mode of the Dirac operator expresses the local properties of the submanifold, such as the Frenet-Serret and generalized Weierstrass relations. In other words this article gives a representation of a further generalized Weierstrass relations for a general k-s...

This paper introduces a submanifold of the moduli space of unitary representations of the fundamental group of a punctured sphere with fixed local monodromy. The submanifold is defined via products of involutions through Lagrangian subspaces. We show that the moduli space of Lagrangian representations is a Lagrangian submanifold of the moduli of unitary representations.

We prove that if the normal distribution of a singular riemannian foliation is integrable, then each leaf of this normal distribution can be extended to be a complete immersed totally geodesic submanifold (called section), which meets every leaf orthogonally. In addition the set of regular points is open and dense in each section. This result generalizes a result of Boualem and solves a problem...

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudoRiemannian space forms. As a corollary, we obtain a non-immersibili...