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I will present an explicit calculation using Chern-Weil theory, which makes an amusing use of Legendre's duplication formula for the gamma function. The Chern character form of a vector bundle $E$ wit …

The answer to this question as asked is no. However, you generally obtain something similar. Consider $D = Q + Q^*$. By standard arguments, $$\ker(D) = \ker(Q)\cap \ker(Q^*).$$ (Of course $D\Phi = 0$ …

Let $M$ be an $4m$-dimensional Riemannian manifold. We can then form the Pontryagin classes $p_k(TM)$ of the tangent bundle using Chern-Weil theory. For any sequence of numbers $k_1, \dots, k_l$ such …

For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function …

The famous maximum principle of Hamilton states the following. Let $C$ be a convex $O(n)$-invariant subset of the space of algebraic curvature operators. Then if it is invariant under the ODE $$ \dot{ …

My question is the following: What is the minimum regularity for a continuous loop $\gamma: S^1 \rightarrow M$ in a Riemannian manifold $M$ to have short-time existence for the harmonic map flow in …

Suppose I have a complete, non-compact Riemannian manifold $M$ such that the volume of balls around a fixed point $p \in M$ satisfies $$\mathrm{vol}(B_R(p)) \leq v(R)$$ for some function $v$. Can we t …

For appropriate choices of $i$, $j$ (e.g. $i+j \neq n$), $\Omega^i(M)$ and $\Omega^j(M)$ have different ranks as $C^\infty(M)$ module, so at least they cannot be isomorphic as modules. Maybe they coul …

In the Wikipedia article on scalar curvature, it is noticed that the Hölder inequality implies $$Y(g) \geq - \left(\int_M |R(g)|^{n/2} \mathrm{d}V_g\right)^{n/2}$$ for the Yamabe functional $Y(g)$ and …

This is probably a longer comment, but let me make this into an answer. Suppose $\gamma$ is a curve in your manifold and we are asking if it satisfies the geodesic equation in some sense. Then either …

Yes, there is indeed a mistake. Chavels Lemma 2 on page 153 tells you that $$L(H_k * F) = (LH_k)*F - F,$$ so if you define $F = \sum_{l=1}^\infty (LH_k)^{*l}$ and $p= H_k + H_k * F$, then $$ L p = LH_ …

Let $M$ be a compact Riemannian manifold and let $V \in C^\infty(M)$. Consider the operator $\Delta + V$ and let $p_t(x, y)$ be the corresponding heat kernel. If $\Delta + V$ is a positive operator su …

In their paper "Conformally flat Manifolds, Kleinian Groups and Scalar Curvature", Schoen and Yau repeatedly use the term "Newtonian capacity" for a subset of $S^n$. I know the following definition: …

Let $M$ be a compact Riemannian manifold and let $\mathrm{d}\mathbb{W}^{yx;T}(\gamma)$ denote the Brownian Bridge measure, i.e. the Wiener measure on the paths that travel from $x$ to $y$ in time $T$ …

Sorry for the confusion from earlier. I tried to fix the thread. The old version can be found below. For $6$-dimensional subspaces $V$ of the space $\mathbb{R}^{3\times 3}$ of real three-times-three …