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Assume that there exists a first order elliptic operator $D$ acting on functions from $\mathbb{R}^n$ to some vector space $V$. What can we conclude about $V$? For example, is the dimension of $V$ alw …

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet s …

I am pretty sure this should be text book material, but I couldn't find this anywhere; maybe I just don't know where to look. Problem: Suppose we have a smooth vector field $X = a_i x^i \partial_i + …

I believe the answer to my question is well-known, but as I do not know too much about harmonic map flows, here it goes: Let $M$ be a complete Riemannian manifold. For a curves $c: [0,1] \longrightar …

The linearization of the vector field $X$ at the singular point zero is $$DX|_0 = \begin{pmatrix} 1 & 1\\ -1 & 0\end{pmatrix},$$ the eigenvalues of which are $$ \lambda_{1, 2} = \frac{1}{2} \pm \frac{ …

Let $\Phi_t(x)$ be the flow of your vector field $\nabla d_{\mathcal{A}}$, starting at $x \in \mathcal{A}$. It will exist until the time $t_0(x)$, when it hits the set where $\nabla d_{\mathcal{A}}$ i …

The result of Poincaré is in fact not about the real case but about the complex case. Suppose a vector field of the form $$ X = \sum_i\lambda_i x^i \partial_i + \text{higher order terms}$$ Then there …

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 …

Since $u$ tends to $0$ as $x$ goes to infinity and it is apparently supposed to be $C^1$, it is bounded. Since the Laplace operator is a negative operator, the operator $L = \Delta - u$ is bounded fro …