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Note: I'm updating my answer to give a better (i.e., simpler) example plus a little more information about how to derive the example from Douglas' results (which may not be entirely clear upon first …

It is known (and follows from an easy calculation) that solutions of the isoperimetric problem on a surface have constant geodesic curvature. In his 1887 classic Leçons Sur La Théorie Générale Des Su …

There is probably no single proof that would provide a rigorous justification of the OP's principle in all cases. Moreover, without specifying more clearly what is meant by a 'natural map', the princ …

Not exactly 'tweetable', but perhaps the identity (1) may help, if all you want to do is avoid the Euler-Lagrange equations. For simplicity, assume that $M^n$ is oriented. (One can write the identit …

If you just calculate using the moving frame, you'll get the answer for the variation of the principal curvatures in a few lines: $$ \delta\kappa_i = \mathrm{Hess}(u)(e_i,e_i) + \kappa_i^2\,u . $$ Her …

There is a coordinate-free description using only natural objects on $TM$. Here is one way to do it. First, consider the basepoint submersion $\pi:TM\to M$. For each $v\in TM$, the linear map $\pi' …

According to Giaquinta and Hildebrandt (Calculus of Variations I, p. 70): "Euler's differential equation was first stated by Euler in his Methodus inveniendi [2], Chapter 2, no. 21. Quite often, one s …

Thanks for explaining your motivation, because I think that the general problem as you stated it is impossibly hard, but that, fortunately, for the problem that you are really trying to tackle (the in …

Now that your comment has clarified your question, we can answer it: The answer is 'no'. There is the following well-known example: Consider the following family of circles: $C_\lambda$ is defin …

There is no reason to believe that there is a supremum of this functional. For example, consider the $3$-torus $M = \mathbb{R}^3/\mathbb{Z}^3$ with the quotient metric and the unit $1$-forms $$ \alph …

Here's an example to show that the infimum is not always attained: Consider the standard Hopf map $\pi:S^3\to S^2$, which is not null-homotopic, of course, so it follows that the area of the graph in …

There is a large literature on this, and the roots go back more than one hundred years. Some of the modern work along these lines can be found by looking for papers containing the term 'variational b …

N.B. This is an edit of my original post, confirming the guess that I made originally. The answer is no, i.e., such curves are not forced to be ellipses. Here is a sketch of the argument. (The det …

Here's a simple counterexample: Let $M=N=T^2$ (the standard torus, thought of as $\mathbb{R}^2/\mathbb{Z}^2$). Let $f:M\to N$ be the identity, and let the metrics on $M$ and $N$ be any two translati …

Actually, your notation is causing some confusion. In one very real sense (probably not your intended one) the answer to your question is yes, not no which is probably the answer to the question that …