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Suppose $S_0,S\subseteq\mathbb{R}^3$ are embedded surfaces. For a smooth map $\phi:S_0\rightarrow S$, define an energy $E[\phi]$ by $$E[\phi]:=\int_{\Sigma_0}F[\Lambda(d\phi_p)]\,dA(p).$$ Here, $\Lambda(d\phi_p)\equiv (\sigma_1(d\phi_p),\sigma_2(d\phi_p))$ is the set of singular values of the Jacobian of $\phi$ at $p\in S_0$. As an example, if we define $F[\sigma_1,\sigma_2]:=\sigma_1^2+\sigma_2^2$, then $E[\phi]$ is the Dirichlet energy of $\phi$.

Here's my question: Given a smooth map $\phi_0:S_0\rightarrow S$, is there a sufficient condition on $F:\mathbb{R}^2\rightarrow\mathbb{R}$ guaranteeing existence of a map $\phi:S_0\rightarrow S$ in the homotopy class of $\phi$ that locally minimizes $E[\cdot]$?


What I have in mind is the gradient flow of Eells and Sampson, which proves existence of harmonic maps when $S$ has negative Gaussian curvature by starting with an arbitrary $\phi_0$ and flowing along the gradient of the Dirichlet energy to a local optimum. This is an elegant construction, but the drawback is that not all target surfaces $S$ admit a harmonic map $\phi:S_0\rightarrow S$.

My intuition is that Eells and Sampson's construction fails in the presence of positive curvature because $F[\sigma_1,\sigma_2]=\sigma_1^2+\sigma_2^2$ "wants to" pinch points, i.e. reach a Jacobian with as small singular values as possible. But perhaps objectives like the symmetric Dirichlet energy, which appears in computer graphics applications, which looks like $F[\sigma_1,\sigma_2]:=\sigma_1^2+\sigma_2^2+\sigma_1^{-2}+\sigma_2^{-2}$, would have better properties since it has an asymptote whenever $\sigma_i=0$ and is minimized when $\sigma_1=\sigma_2=1$.

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    $\begingroup$ I think it's very difficult - as soon as you choose a different energy you're dealing with a quasilinear system, and very few of the estimates from HMHF carry over. I've been trying the Hölder approach to a similar problem (looking for harmonic diffeomorphisms using a gradient-like flow for the Dirichlet energy) for a long while now and have only managed to get it working in the absence of curvature - see arxiv.org/abs/1609.08317. It's possible the gradient flow structure could make things easier, but it certainly won't be as easy as Eells-Sampson. $\endgroup$ – Anthony Carapetis Apr 1 '17 at 10:08
  • $\begingroup$ Hmm, that's too bad! I was hoping somehow things would be easier than Eells-Sampson if we build an objective function $F$ that resists collapse into singularities (e.g. the symmetric Dirichlet energy above, which has an asymptote that "wants to" avoid singular Jacobians). This seems to be the empirical observation in the applied world but we don't have math to back it up! I'll take a look at your paper ---- if you're curious about the applications we're considering, shoot me an email and I'd be happy to sketch out details! $\endgroup$ – Justin Solomon Apr 1 '17 at 13:26

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