In this paper we prove the path connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four-dimensional manifolds.
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New proof shows path-connectedness of actions on intervals and circles.
problem Path-connectedness of actions of .
method New proof using diffeomorphisms with absolutely continuous derivative.
result Path-connectedness of the space of actions.
Study on Frechet distance properties for paths and graphs.
problem Understanding topological properties of Frechet distance spaces.
method Proving path-connectedness of Frechet distance spaces and metric balls.
result Spaces of paths and graphs under Frechet distance are path-connected.
We prove that the moduli space of 2-convex embedded n-spheres in R^{n+1} is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the path-connectedness of the moduli space of positive scalar curvature metics on three-manifold…
In this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact 3-manifold is path-connected. The proof uses the Ricci flow with surgery, the conformal method, and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton's Ricci flow is fundam…
Proves new inequality linking spectral numbers of Lagrangians and their reductions.
problem Understanding spectral properties of Lagrangian submanifolds.
method Develops inverse reduction inequalities for spectral numbers.
result Proof of inequality between spectral numbers of Lagrangian and its reductions.
A 3D space of hyperbolic manifolds is connected but not path-connected.
problem Proving connectivity and non-path-connectedness of framed hyperbolic 3-manifolds.
method Two proofs using density theorems for Kleinian groups, constructing dense sets of framings, and discussing paths.
result The space of framed infinite volume hyperbolic 3-manifolds is not path-connected.
The study examines mean curvature flow and Heegaard surfaces in lens spaces.
problem Analyzing mean convex two-spheres and Heegaard tori in lens spaces.
method Proves path-connectedness of moduli spaces of mean convex two-spheres and Heegaard tori in manifolds with nonnegative Ricci curvature.
result There are always either one or two path components of mean convex Heegaard tori in lens spaces, depending on the ambient manifold's homotopy type.
The paper proves parabolic gap theorems for Yang-Mills energy.
problem Yang-Mills energy and instantons on various manifolds.
method Parabolic Yang-Mills flow and Morrey norms.
result Spaces of connections with Yang-Mills energy less than a certain threshold deformation-retract onto spaces of instantons.
Based on a novel type of Sobolev-Poincaré inequality (for generalised weakly differentiable functions on varifolds), we establish a finite upper bound of the geodesic diameter of generalised compact connected surfaces-with-boundary of arbitrary dimension in Euclidean space in terms of the mean curvatures of the surface…
Study metrics with specific spectral properties to compute Bartnik and Bartnik-Bray masses efficiently.
problem Compute Bartnik and Bartnik-Bray masses efficiently for metrics with specific spectral properties.
method Spectral generalization of positive scalar curvature, applying Codá Marques's path-connectedness theorem, and efficient constructions for scalar-nonnegative fill-in problem.
result Compute Bartnik and Bartnik-Bray masses efficiently for metrics with -Δ + kR ≥ 0.
On the Moduli Space of Asymptotically Flat Manifolds with Boundary and the Constraint Equationsmath.DG
Let be a closed -manifold, the space of metrics on with positive scalar curvature, and the group of diffeomorphisms of . Marques proves the fundamental result that is path connected. Using this and the theorem of Cerf in differential…
For a connected locally path-connected topological space and a continuous function on it such that its Reeb graph is a finite topological graph, we show that the cycle rank of , i.e., the first Betti number , in computational geometry called \emph{number of loops}, is bounded from above by …