In , R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator.
We show that this inequality holds under the weaker condition that M × R2 has nonnegative isotropic curvature.
Given an SO(3)-bundle with connection, the associated twosphere
bundle carries a natural closed 2-form. Asking that this be
symplectic gives a curvature inequality first considered by Reznikov
. We study this inequality in the case when the base has
dimension four, with three main aims.
Firstly, we use this approach to construct symplectic six-manifolds
with c1 = 0 which are never K¨ahler; e.g., we produce such
manifolds with b1 = 0 = b3 and also with c2 · [ω] < 0, answering
questions posed by Smith–Thomas–Yau .
Examples come from Riemannian geometry, via the Levi–Civita
connection on +. The underlying six-manifold is then the twistor
space and often the symplectic structure tames the Eells–Salamon
twistor almost complex structure. Our second aim is to exploit
this to deduce new results about minimal surfaces: if a certain
curvature inequality holds, it follows that the space of minimal
surfaces (with fixed topological invariants) is compactifiable; the
minimal surfaces must also satisfy an adjunction inequality, unifying
and generalising results of Chen–Tian .
One metric satisfying the curvature inequality is hyperbolic
four-space H4. Our final aim is to show that the corresponding
symplectic manifold is symplectomorphic to the small resolution
of the conifold xw −yz = 0 in C4. We explain how this fits into a
hyperbolic description of the conifold transition, with isometries of
H4 acting symplectomorphically on the resolution and isometries
of H3 acting biholomorphically on the smoothing.
The diameter of a disc filling a loop in the universal covering of a Riemannian manifold M may be measured extrinsically using the
distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter
of a van Kampen diagram filling a word that represents the identity in a finitely presented group can either be measured intrinsically in the 1–skeleton of or extrinsically in the Cayley graph of . We construct the first examples of closed manifolds M and finitely presented groups = π1M for which this choice — intrinsic versus extrinsic — gives rise to qualitatively different min–diameter filling functions.
Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) al-
most normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following results:
a minimal triangulation of a closed irreducible or a boundedhyperbolic 3-manifold contains no non-trivial k-normal sphere;
every triangulation of a closed manifold with at least 2 tetra-hedra contains some non-trivial normal surface;
every manifold with boundary has only finitely many triangulations without non-trivial normal surfaces. Here, triangulations of bounded manifolds are actually ideal triangulations. We also calculate the number of normal surfaces of nonnegative Euler characteristics which are contained in the conjecturally minimal triangulations of all lens spaces Lp,q.
Let w be an Abelian differential on a compact Riemann surface of genus g ≥ 1. Then |w|2 defines a flat metric with conical singularities
and trivial holonomy on the Riemann surface. We obtain an explicit holomorphic factorization formula for the ζ-regularized determinant of the Laplacian in the metric |w|2, generalizing the classical Ray-Singer result in g = 1.