Hilbert Manifold Research Articles

Gradient Flow of the Sinai–Ruelle–Bowen Entropy

Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow’s ordinary differential equation representation. This gradient flow has close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation.

Global solution of nonlinear Heat equation with solutions in a Hilbert Manifold

The objective of this paper is to deal with the deterministic problem consisting of non-linear heat equation of gradient type. The evolution equation emerges as projecting the Laplace operator with Dirichlet boundary conditions and polynomial nonlinearity of degree 2n−1, onto the tangent space of a sphere M in a Hilbert space ℋ. We are going to deal with questions of the existence and the uniqueness of a global solution, and the invariance of the manifold, M i.e. if the suitable initial data lives on M then all trajectories of solutions also belong to M. Finally, we will also show the global solution generates a gradient flow.

Low-regularity Seiberg–Witten moduli spaces on manifolds with boundary

For a compact spin^{c} manifold X with boundary b_1(partial X)=0, we consider moduli spaces of solutions to the Seiberg–Witten equations in a generalized double Coulomb slice in W^{1,2} Sobolev regularity. We prove they are Hilbert manifolds, prove denseness and “semi-infinite-dimensionality” properties of the restriction to partial X, and establish a gluing theorem. To achieve these, we prove a general regularity theorem and a strong unique continuation principle for Dirac operators, and smoothness of a restriction map to configurations of higher regularity on the interior, all of which are of independent interest.

Faedo-Galerkin Approximations for nonlinear Heat equation on Hilbert Manifold

In this work, we aim to study the well-posedness of a deterministic problem consisting of the non-linear heat equation of gradient type. The evolution equation emerges by projecting the Laplace operator with Dirichlet boundary conditions and polynomial nonlinearity of degree $2n-1$, onto the tangent space of the sphere $\mathcal{M}$ in a Hilbert space $\mathcal{H}$. We are going to deal with the question of existence and uniqueness based on the Faedo-Galerkin compactness method.

Bartnik–Hilbert manifold structure on fibers of the scalar curvature and the constraint operator

Bartnik–Hilbert manifold structure on fibers of the scalar curvature and the constraint operator

Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblems

We propose to use the Łojasiewicz inequality as a general tool for analyzing the convergence rate of gradient descent on a Hilbert manifold, without resorting to the continuous gradient flow. Using this tool, we show that a Sobolev gradient descent method with adaptive inner product converges exponentially fast to the ground state for the Gross-Pitaevskii eigenproblem. This method can be extended to a class of general high-degree optimizations or nonlinear eigenproblems under certain conditions. We demonstrate this generalization by several examples, in particular a nonlinear Schrodinger eigenproblem with an extra high-order interaction term. Numerical experiments are presented for these problems.

An SQP Method for Equality Constrained Optimization on Hilbert Manifolds

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 27 May 2020Accepted: 18 May 2021Published online: 16 September 2021Keywordsequality constrained optimization, variational problems, optimization on Hilbert manifoldsAMS Subject Headings49M37, 90C55, 90C06Publication DataISSN (print): 1052-6234ISSN (online): 1095-7189Publisher: Society for Industrial and Applied MathematicsCODEN: sjope8

Hilbert manifold structure for weakly asymptotically hyperbolic relativistic initial data

We provide a Hilbert manifold structure a la Bartnik for the space of asymptotically hyperbolic initial data for the vacuum constraint equations. The adaptation led us to prove new weighted Poincare and Korn type inequalities for AH manifolds with inner boundary and weakly regular metric.

Variational Obstacle Avoidance with Applications to Interpolation Problems in Hybrid Systems

Abstract We study variational obstacle avoidance problems on complete Riemannian manifolds and apply the results to the construction of piecewise smooth curves interpolating a set of knot points in systems with impulse effects. We derive the dynamical equations for extrema in the variational problem, and show the existence of minimizers by using lower-continuity arguments for weak convergence on an infinite-dimensional Hilbert manifold. We then provide conditions under which it is possible to ensure that the extrema will safely avoid a given obstacle within some desired tolerance.

Global solution of nonlinear stochastic heat equation with solutions in a Hilbert manifold

The objective of this paper is to prove the existence of a global solution to a certain stochastic partial differential equation subject to the [Formula: see text]-norm being constrained. The corresponding evolution equation can be seen as the projection of the unconstrained problem onto the tangent space of the unit sphere [Formula: see text] in a Hilbert space [Formula: see text].

Weil–Petersson Teichmüller space III: dependence of Riemann mappings for Weil–Petersson curves

The classical Riemann mapping theorem implies that there exists a so-called Riemann mapping which takes the upper half plane onto the left domain bounded by a Jordan curve in the extended complex plane. The primary purpose of the paper is to study the basic problem: how does a Riemann mapping depend on the corresponding Jordan curve? We are mainly concerned with those Jordan curves in the Weil–Petersson class, namely, the corresponding Riemann mappings can be quasiconformally extended to the whole plane with Beltrami coefficients being square integrable under the Poincare metric. After giving a geometric characterization of a Weil–Petersson curve, we endow the space of all normalized Weil–Petersson curves with a new real Hilbert manifold structure in a geometric manner and show that this new structure is topologically equivalent to the standard complex Hilbert manifold structure, which implies that an appropriately chosen Riemann mapping depends continuously on a Weil–Petersson curve (and vice versa). This can be considered as the first result about the continuous dependence of Riemann mappings on non-smooth Jordan curves.

Weil-Petersson Teichmüller space revisited

In our previous paper [23] it was proved that a sense-preserving homeomorphism g on the unit circle S1 belongs to the Weil-Petersson class WP(S1), namely, g can be extended to a quasiconformal mapping to the unit disk whose Beltrami coefficient is square integrable in the Poincaré metric if and only if g is absolutely continuous such that log⁡g′ belongs to the Sobolev class H12. In this sequel to [23], we show that the smooth Hilbert manifold structure on WP(S1) inherited from H12 by the pullback g↦log⁡|g′| is compatible with the standard Hilbert manifold structure introduced by Takhtajan-Teo [27]. This enables us to give a fast approach to some results in our previous papers [23] and [24].

A remark on vanishing geodesic distances in infinite dimensions

We observe that a vanishing geodesic distance arising from a weak Riemannian metric in a Hilbert manifold can be constructed.

Nonholonomic and constrained variational mechanics

Equations governing mechanical systems with nonholonomic constraints can be developed in two ways: (1) using the physical principles of Newtonian mechanics; (2) using a constrained variational principle. Generally, the two sets of resulting equations are not equivalent. While mechanics arises from the first of these methods, sub-Riemannian geometry is a special case of the second. Thus both sets of equations are of independent interest.The equations in both cases are carefully derived using a novel Sobolev analysis where infinite-dimensional Hilbert manifolds are replaced with infinite-dimensional Hilbert spaces for the purposes of analysis. A useful representation of these equations is given using the so-called constrained connection derived from the system's Riemannian metric, and the constraint distribution and its orthogonal complement. In the special case of sub-Riemannian geometry, some observations are made about the affine connection formulation of the equations for extremals.Using the affine connection formulation of the equations, the physical and variational equations are compared and conditions are given that characterise when all physical solutions arise as extremals in the variational formulation. The characterisation is complete in the real analytic case, while in the smooth case a locally constant rank assumption must be made. The main construction is that of the largest affine subbundle variety of a subbundle that is invariant under the flow of an affine vector field on the total space of a vector bundle.

On Hilbert–Hartogs manifolds

We prove some extension results for holomorphic mappings with values in complex Hilbert manifolds.

Weil-Petersson Teichmüller space II: Smoothness of flow curves of [formula omitted]-vector fields

Given a continuous vector field λ(t,⋅) of Sobolev class H32 on the unit circle S1, the flow maps η=g(t,⋅) of the differential equation{dηdt=λ(t,η)η(0,ζ)=ζ are known to be quasisymmetric homeomorphisms. Very recently, Gay-Balmaz-Ratiu [15] conjectured that the flow curve g(t,⋅) is in the Weil-Petersson class WP(S1) and is continuously differentiable with respect to the Hilbert manifold structure of WP(S1) introduced by Takhtajan-Teo [40]. The first assertion had already been demonstrated in our previous paper [36]. In this sequel to [36], we will continue to deal with the Weil-Petersson class WP(S1) and completely solve this conjecture in the affirmative.

Smooth approximations without critical points of continuous mappings between Banach spaces, and diffeomorphic extractions of sets

Let E, F be separable Hilbert spaces, and assume that E is infinite-dimensional. We show that for every continuous mapping f:E→F and every continuous function ε:E→(0,∞) there exists a C∞ mapping g:E→F such that ‖f(x)−g(x)‖≤ε(x) and Dg(x):E→F is a surjective linear operator for every x∈E. We also provide a version of this result where E can be replaced with a Banach space from a large class (including all the classical spaces with smooth norms, such as c0, ℓp or Lp, 1<p<∞), and F can be taken to be any Banach space such that there exists a bounded linear operator from E onto F. In particular, for such E,F, every continuous mapping f:E→F can be uniformly approximated by smooth open mappings. Part of the proof provides results of independent interest that improve some known theorems about diffeomorphic extractions of closed sets from Banach spaces or Hilbert manifolds.

Infinite-dimensional Log-Determinant divergences between positive definite Hilbert–Schmidt operators

The current work generalizes the author’s previous work on the infinite-dimensional Alpha Log-Determinant (Log-Det) divergences and Alpha-Beta Log-Det divergences, defined on the set of positive definite unitized trace class operators on a Hilbert space, to the entire Hilbert manifold of positive definite unitized Hilbert–Schmidt operators. This generalization is carried out via the introduction of the extended Hilbert–Carleman determinant for unitized Hilbert–Schmidt operators, in addition to the previously introduced extended Fredholm determinant for unitized trace class operators. The resulting parametrized family of Alpha-Beta Log-Det divergences is general and contains many divergences between positive definite unitized Hilbert–Schmidt operators as special cases, including the infinite-dimensional affine-invariant Riemannian distance and the infinite-dimensional generalization of the symmetric Stein divergence.

Lower Semi-continuity of the Index in the Viscosity Method for Minimal Surfaces

AbstractThe goal of the present work is two-fold. First we prove the existence of a Hilbert manifold structure on the space of immersed oriented closed surfaces with three derivatives in $L^2$ in an arbitrary compact submanifold $M^m$ of an Euclidian space ${{\mathbb{R}}}^Q$. Second, using this Hilbert manifold structure, we prove a lower semi-continuity property of the index for sequences of conformal immersions, critical points to the viscous approximation of the area satisfying a Struwe entropy estimate and a bubble tree strongly converging in $W^{1,2}$ to a limiting minimal surface as the viscous parameter is going to zero.

On the homotopy classification of proper Fredholm maps into a Hilbert space

Abstract We classify the homotopy classes of proper Fredholm maps from an infinite-dimensional Hilbert manifold into its model space in terms of a suitable version of framed cobordism. Our construction is an alternative approach to the classification introduced by Elworthy and Tromba in 1970 and does not make use of further structures on the ambient manifold, such as Fredholm structures. In the special case of index zero, we obtain a complete classification involving the Caccioppoli–Smale mod 2 degree and the absolute value of the oriented degree.