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Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean ...
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Differentiable structure on the gauge group?

In this paper I have come across a formulation involving differentiation in the gauge group of a principal bundle which I do not understand (found at the very top of p. 369). Let $P\rightarrow M$ be ...
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73 views

Vector space structure on $T_pM$ again

This minor problem popped up while I was reading the book "Modern Differential Geometry for Physicists" by Chris J. Isham. It deals with introducing vector space structure on a tangent space $T_pM$ to ...
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Constant such that Global Maximum is Particular Entry in Vector

Suppose I have $N\times 1$ vectors $X$ and $Y$, such that they are both are strictly increasing. I define a third vector as $Z\equiv f(X) + K g(Y)$ where $f$ and $g$ are arbitrary functions. I want ...
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32 views

Banach manifold structure for sets of maps

I am looking at the lecture note of J.D.Moore, which is available at "www.math.ucsb.edu/~moore/globalanalysisshort.pdf". At page 16-17(just before Lemma 1.3.1), it explains the method to endow ...
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Few questions about global analysis relating $C^k$ functions

First question is about the definition. Let $U$ be an open subset of $R^n$. Let $f$ be $k$ times continuously differentiable function on $U$. $C^k$ norm of $f$ is defined as sum of uniform norm of ...
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42 views

Integral kernel of composition of two operators

Let $T_1$ and $T_2$ be two operators on $L^2(M)$ where $M$ is a Riemannian manifold, with integral kernels $k_1(x,y)$ and $k_2(x, y)$ respectively. I am trying to derive some estimate on $$\int_B k(x, ...
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39 views

$C^{0}$-norm of a map

I have the following question: Consider a continous map $f:M\rightarrow N$, where $M,N$ are smooth manifolds. How can one define its $C^{0}(M,N)$-norm, i.e. $||f||_{C^{0}(M,N)}$ ? Greetings, Daniel