Stephanie B. Alexander

Geometric curvature bounds in Riemannian manifolds with boundary

An Alexandrov upper bound on curvature for a Riemannian manifold with boundary is proved to be the same as an upper bound on sectional curvature of interior sections and of sections of the boundary which bend away from the interior. As corollaries those same sectional curvatures are related to estimates for convexity and conjugate radii; the Hadamard-Cartan theorem and Yaus isoperimetric inequality for spaces with negative curvature are generalized

Comparison theorems for curves of bounded geodesic curvature in metric spaces of curvature bounded above

Comparison and rigidity theorems are proved for curves of bounded geodesic curvature in singular spaes of curvature bounded above. Most of these estimates do not appear in the literature even for smooth curves in Riemannian manifolds. Geodesic curvature (which agrees with the usual one in the smooth case) is defined by comparison to curves of constant curvature in a model space. Two methods of comparison are used, preserving either sidelengths of inscribed triangles or arclength and chordlength. Using a majorization theorem of Reshetnyak, we obtain best possible global comparisons for arclength, chordlength, width and base angles in a CAT(K) space. A criterion for a metric ball to be a CAT(K) space is also given, in terms of the radius and the radial uniqueness property.

Capture pursuit games on unbounded domains

We introduce simple tools from geometric convexity to analyze capturetype (or “Lion and Man”) pursuit problems in unbounded domains. The main result is a necessary and sufficient condition for eventual capture in equal-speed discrete-time multi-pursuer capture games on convex Euclidean domains of arbitrary dimension and shape. This condition is presented in terms of recession sets in unit tangent spheres. The chief difficulties lie in utilizing the boundary of the domain as a constraint on the evader’s escape route. We also show that these convex-geometric techniques provide sufficient criteria for pursuit problems in non-convex domains with a convex decomposition.

The convex hull property and topology of hypersurfaces with nonnegative curvature

Abstract We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed hypersurface lies outside the convex hull of its boundary, provided the boundary satisfies certain required conditions. This gives a convex hull property, dual to the classical one for surfaces with nonpositive curvature. A version of this result in the nonsmooth category is obtained as well. We show that our boundary conditions determine the topology of the surface up to at most two choices. The proof is based on uniform estimates for radii of convexity of these surfaces under a clipping procedure, a noncollapsing convergence theorem, and a gluing procedure.

The Fary-Milnor theorem in Hadamard manifolds

The Fary-Milnor theorem is generalized: Let γ be a simple closed curve in a complete simply connected Riemannian 3-manifold of nonpositive sectional curvature. If γ has total curvature less than or equal to 4π, then γ is the boundary of an embedded disk. The example of a trefoil knot which moves back and forth abritrarily close to a geodesic segment shows that the bound 4π is sharp in any such space. The original theorem was for closed curves in Euclidean 3-space and the proof by integral geometry did not apply to spaces of variable curvature. Now, instead, a combinatorial proof has been devised.

Topology of Riemannian submanifolds with prescribed boundary

We prove that a smooth compact submanifold of codimension 2 immersed in R, n ≥ 3, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high, or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash’s isometric embedding theorem, and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman.

Spines and topology of thin Riemannian manifolds

Consider Riemannian manifolds M for which the sectional curvature of M and second fundamental form of the boundary B are bounded above by one in absolute value. Previously we proved that if M has sufficiently small inradius (i.e. all points are sufficiently close to the boundary), then the cut locus of B exhibits canonical branching behavior of arbitrarily low branching number. In particular, if M is thin in the sense that its inradius is less than a certain universal constant (known to lie between.108 and.203), then M collapses to a triply branched simple polyhedral spine. We use a graphical representation of the stratification structure of such a collapse, and relate numerical invariants of the graph to topological invariants of M when B is simply connected. In particular, the number of connected strata of the cut locus is a topological invariant. When M is 3-dimensional and compact, M has complexity 0 in the sense of Matveev, and is a connected sum of p copies of the real projective space P 3 , t copies chosen from the lens spaces L(3,±1), and l handles chosen from S 2 x S 1 or S 2 XS 1 , with β 3-balls removed, where p + t + l + β > 2. Moreover, we construct a thin metric for every graph, and hence for every homeomorphism type on the list.

NONNEGATIVELY CURVED ENDS OF EUCLIDEAN HYPERSURFACES

The structure of the ends of immersed Euclidean hypersurfaces is described, assuming curvature positive outside a compact subset, or curvature nonnegative and nullity of the second fundamental form at most one outside a compact subset. Examples show the results are optimal.

Racetracks and Extermal Problems for Curves of Bounded Curvature in Metric Spaces

For a closed curve in a CAT(K) space with given circumradius and upper bound on curvature, a basic lower bound on the length is established. The inequality is sharp, assumed only when the curve is the boundary of an isometric copy of a racetrack (the convex hull of two congruent circles) from a plane of constant curvature K. Previously such a theorem was proved for Euclidean plane curves by G.D.Chakerian, H.H. Johnson, and A. Vogt, and for curves in higher dimensional Euclidean spaces by A.D. Milka. A similar theorem is proved for nonclosed curves, with a notion of breadth replacing circumradius. Thus we illustrate how singular methods can extend classical Euclidean theorems to a large class of new spaces (including Riemannian manifolds of curvature bounded above) and also give significant strengthenings even in Euclidean space.