1
vote
1answer
68 views

Gradient of norm of embedding

Let $\varphi:(M,g,\nabla)\to\mathbb{R}^n$ be a smooth embedding of a convex hypersurface. I want to explicitly calculate $$\langle \varphi,\varphi_{\ast}(\nabla\|\varphi\|^2)\rangle.$$ In particular, ...
0
votes
0answers
4 views

Perturbation of the boundary of a strictly stable minimal surface

Let $\Sigma \subseteq \mathbb{R}^3$ be a minimal surface with boundary $\Gamma$. Now let us assume that $\Sigma$ is strictly stable, that is, $\lambda_1(\Sigma,L) >0$, where $L$ is the stability ...
2
votes
2answers
48 views

Are all critical points of energy geodesics?

Let $\gamma$ be a smooth curve in a Riemannian manifold and consider the arclength functional $L(\gamma) = \int_a^b |\gamma'(t)|\, dt$ and the energy functional $E(\gamma) = \frac{1}{2}\int_a^b ...
3
votes
1answer
159 views

Euler Lagrange equation for harmonic maps

In the paper "The existence of minimal immersions of 2-spheres" by Sacks and Uhlenbeck the authors claim that the Euler Lagrange equation for the modified functional $E_\alpha(s) = \int_M (1 + ...
2
votes
1answer
58 views

Is a geodesic the least curved path?

It is clear that, in $\mathbb{R}^n$, straight lines are the lines with minimum possible curvature. That is, given the Frenet-Serret ($n$-dimensional equivalent) matrix, and taking its squared norm, ...
1
vote
0answers
33 views

Minimum is attained in a subset of a Sobolev space

Let $\Omega \subset \mathbb R^n$. I have a functional of the form, $$\int_{\Omega}f(x,u,\nabla u)dx$$ where $u \in W^{1,p}(\Omega, M)$, $M \subset \mathbb R^d$ is a compact, smooth Riemannian ...
3
votes
1answer
88 views

Geodesics in a manifold M diffeomorphic to $\mathbb S^2$

I am now reading the book Calculus of Variations written by Jost and I encountered the following problem (in Theorem 2.3.3.): Let $M$ be a differentiable submanifold of $\mathbb R^d$ diffeomorphic to ...
0
votes
1answer
75 views

Simplifying the search for a geodesic

How can calculations for the geodesics on the surface $U=\{(x,y,z): c(x^2+y^2)-z^2=0, z>0\}$ be simplified by noting that is locally Euclidean? I can see that the property means that when we open ...