Examples of rough metrics

[pabryan]

It be very helpful to describe some examples, particularly in connection with the heat kernel estimates.

• Euclidean space. Here Tychonoff's example shows non-uniqueness of solutions of the heat equation! Such examples are not $L^2$. Effectively, the maximum principle cannot be applied. If one restricts to controlled growth like $e^{d^2}$ at $\infty$, then the max principle and uniqueness are restored via localisation techniques. Our setup works for initially $L^2$ data which is more restrictive. This also would serve as a nice illustration of how we phrase the max principle as non-negative heat kernel (non-negative preserving) while the strong maximum principle (which is often proved using the Harnack inequality) as positive heat kernel (positive preserving). An immediate corollary is uniqueness of solutions to the heat equation which we do get here, provided we work say in $L^2$.
• Round cones should be measurably isometric with Euclidean space. Do heat kernel estimates for rough metrics see cone singularities? Would be a nice illustrative example.
• Rectifiable sets and particularly non-smooth minimal surfaces. These are countable unions of Lipschitz images of open sets of Euclidean space, or even better, $C^1$ maps apart from a set of small measure: see "Rectifiable Set" at Wikipedia. In particular, do our results apply to non-smooth minimal surfaces. This would have considerable interest.
• Conical singularities such as warped products and particularly Kahler Einstein metrics with conical singularities that arise as limits
• orbifolds? lifting a metric from an orbifold to the smooth cover gives a rough metric?
• applications to limits of manifolds/metrics and blow ups. Under Ricci flow can one obtain singular limits are rough metrics? See Perelman perhaps? Local Euclidean comparability via isoperimetric bounds? What about injectivity radius and curvature bounds? Compactness and Sobolev embedding here? This would have considerable interest
• connection with RCD spaces via Saloff-Coste

[lashputin]

• We can't do anything away from L^2, our methods are very much L^2. This is the price we have to pay in order to be able to build this stuff at such generality. In fact, how do we even define any of these objects away from L^2? we could look at it distributionally, but I can't see how a distributional Laplacian would have a heat kernel?

• I'm not sure of what a round cone is? Do you mean a cone that has S^n cross sections? A manifold with such a cone attached is indeed a smooth manifold with a rough metric. This is in my paper with Mike and Sajjad.

• I would very much doubt anything rectifiable can be realised as a manifold with rough metric in general. We know that in dimensions > 3, not every Lipschitz manifold can be smoothed. This indicates that it would be impossible in general to pass topological singularities to those in a metric, because one would first need to smooth out the structure of the Lipschitz manifold, which is not always possible.

• A general warped product dr^2 + g_N with cross section N will not be a rough metric, because the singularity at r = 0 might be topological. If N = S^n, we're all good.

• Not sure about orbifolds at all, but this seems interesting.

• Not sure about this stuff at all. I don't think anyone has proved any kind of limit or compactness theorems in the rough metric setting.

• We can't talk about RCD for rough metrics because the whole issue is that we don't have a metric structure. BUt you make a point here. Maybe if there was a distance associated to the heat kernel of a rough metric, i.e., the limit lim_{t->0} t log( \rho_t(x,y)) exists and gives rise to a metric, then one can ask what conditions on the rough metric would make (M, d, \mu) an RCD-space. I think we're far from this, but.