Let $(M^{n},g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity required so that the solution to a free Schrödinger equation converges pointwise to its initial data. Assume the initial data is in $H^{\unicode[STIX]{x1D6FC}}(M)$. For hyperbolic space, the standard sphere, and the two-dimensional torus, we prove that $\unicode[STIX]{x1D6FC}>\frac{1}{2}$ is enough. For general compact manifolds, due to the lack of a local smoothing effect, it is hard to improve on the bound $\unicode[STIX]{x1D6FC}>1$ from interpolation. We managed to go below 1 for dimension ${\leqslant}$ 3. The more interesting thing is that, for a one-dimensional compact manifold, $\unicode[STIX]{x1D6FC}>\frac{1}{3}$ is sufficient.

Motivated by Almgren’s work on the isoperimetric inequality, we prove a sharp inequality relating the length and maximum curvature of a closed curve in a complete, simply connected manifold of sectional curvature at most −1. Moreover, if equality holds, then the norm of the geodesic curvature is constant and the torsion vanishes. The proof involves an application of the maximum principle to a function defined on pairs of points.

(Local) self-similarity is a seminal concept, especially for Euclidean random fields. Westudy in this paper the extension of these notions to manifold indexed fields. We giveconditions on the (local) self-similarity index that ensure the existence of fractionalfields. Moreover, we explain how to identify the self-similar index. We describe a way ofsimulating Gaussian fractional fields.

Let $(M,g)$ be a manifold of bounded geometry with metric $g$. We consider a Schrödinger-type differential expression $H=\Delta_M+V$, where $\Delta_M$ is the scalar Laplacian on $M$ and $V$ is a non-negative locally integrable function on $M$. We give a sufficient condition for $H$ to have an $m$-accretive realization in the space $L^1(M)$.špace{-4pt}

We show that diagrammatically reducible two-complexes are characterized by the property: every finity subconmplex of the universal cover collapses to a one-complex. We use this to show that a compact orientable three-manifold with nonempty boundary is Haken if and only if it has a diagrammatically reducible spine. We also formulate an nanlogue of diagrammatic reducibility for higher dimensional complexes. Like Haken three-manifolds, we observe that if n ≥ 4 and M is compact connected n-dimensional manifold with a traingulation, or a spine, satisfying this property, then the interior of the universal cover of M is homeomorphic to Euclidean n-space.

We give necessary and sufficient conditions for a norm-compact subset of a Hilbert space to admit a ${{C}^{1}}$ embedding into a finite dimensional Euclidean space. Using quasibundles, we prove a structure theorem saying that the stratum of $n$-dimensional points is contained in an $n$-dimensional ${{C}^{1}}$ submanifold of the ambient Hilbert space. This work sharpens and extends earlier results of $\text{G}$. Glaeser on paratingents. As byproducts we obtain smoothing theorems for compact subsets of Hilbert space and disjunction theorems for locally compact subsets of Euclidean space.

In this paper we consider the well-known implicit Lagrange problem: find a trajectory solution of an underdetermined implicit differentialequation, satisfying some boundary conditions and which is a minimum of the integral of a Lagrangian. In the tangent bundle of the surroundingmanifold X, we define the geometric framework of q-pi- submanifold. This is an extension of the geometric framework of pi- submanifold,defined by Rabier and Rheinboldt for determined implicit differential equations, to underdetermined implicit differential equations. With thisgeometric framework we define a class of well-posed implicit differential equations for which we locally obtain, by means of a reductionprocedure, a controlled vector field on a submanifold W of the surrounding manifold X. We then show that the implicit Lagrange problem leadsto, locally, an explicit optimal control problem on the submanifold W for which the Pontryagin maximum principle is naturally used.