Thanks, I'd seen that when posting but hadn't realized that the second question there is equivalent to the second question here

BranimirĆaćić At the moment I'm not so interested in the proof, I just want to properly understand the meaning of the topological terms in the formula. So what I understand now is that there's three interpretations- one via k-theory on the thom space, one via Quillen superconnections, and another as an asymptotic term in a solution of the heat equation. I like both of the latter two, but the first is still somewhat mysterious to me. So I suppose what I'm most interested in is understanding the K-theoretic viewpoint in 'smooth' terms, something like what @S.carmeli suggests

Is Stein's proof really capable of recovering Schoen-Simon-Yau's in the general case of a Riemannian manifold? I don't see how one would incorporate the Codazzi equation, when the ambient curvature is nonzero, into it

Maybe I had just incorrectly understood the nature of the heat kernel proof. So does the heat kernel proof offer a simplification beyond the use of Quillen superconnections, as in the last paragraph of my question?

I see... I suppose that does answer my question as I phrased it (in the case of Dirac operators), but I was hoping for a direct prescription, via curvature polynomials or such. If I understand correctly, the heat kernel proof determines the form by more transcendental methods, something like as a term in the asymptotic expansion of a solution of a differential equation?

@GabeK the Li-Yau inequality would be an example of all four criteria, not just the first three, wouldn't it? From scalar equations to Ricci flow and mean curvature flow, and also including the original Yau and Cheng-Yau gradient estimates in the elliptic setting

Pogorelov showed the following more general result. For a general metric on $S^2$, if the Gaussian curvature is strictly greater than $-\kappa$ for a nonnegative number $\kappa$, then the metric can be isometrically embedded into the simply-connected complete space form of curvature $-\kappa$.

In chapter 6 of Kobayashi-Nomizu "Foundations of Differential Geometry vol. 1" these are covered under the keyword "affine mapping"

Thanks, I should have remembered your first point... so any good application would have to come with some precise understanding of the extrinsic distance function? The second point is certainly interesting, but is it so crucial? As far as I know the Hamilton-Perelman analysis didn't need to know anything about stability of singularities

I mean the group $\text{O}(V^{\otimes n},g_{\otimes n})$