Let us consider, a closed Riemannian surface $(\Sigma,h)$ and a compact Riemannian manifold $(N,g)$ with dimension greater than $3$. If we are given a sequence of harmonic maps $u_n:(\Sigma,h) \rightarrow (N,g)$ with bounded energy, i.e. $$E(u_n)=\int_\Sigma \vert du_n\vert^2 \, dv < C,$$ it is well know that we have an energy identity, that is to say there exists an harmonic map $u^\infty : (\Sigma,h) \rightarrow (N,g)$ and some bubbles, i.e. harmonic maps $\omega_i:\mathbb{C} \rightarrow (N,g)$ such that

$$\lim_n E(u_n)= E(u^\infty)+\sum_{i} E(\omega_i).$$ My question is:

Does $i\geq 1$ really occurs?

When $\Sigma=\hat{\mathbb{C}}$, the answer is clearly yes, considering $$u_n(z)=(z,nz).$$ But here, the fact that the conformal group of $\hat{\mathbb{C}}$ is not compact seems to be crucial. So is there an example of bubbling when $\Sigma$ is not $\mathbb{C}$, especially in genus bigger than $2$?


Yes. The genus of $\Sigma$ is not really relevant. Here's an example: Let $f$ and $g$ be two meromorphic functions on $\Sigma$, where $g$ is nonconstant, and consider the sequence of maps $u_n: \Sigma\to N^4 = \mathbb{CP}^1\times\mathbb{CP}^1$ given by $$ u_n(p) = \bigl([f(p)],[n\,g(p)]\bigr). $$ (Here, $N$ is given the product metric and the metric on $\mathbb{CP}^1$ is the standard metric of constant sectional curvature $1$.) As $n$ goes to $\infty$, the energy densities of these (holomorphic and, hence, harmonic) maps stay bounded away from the zero divisor of $g$, but go to infinity in a neighborhood. In the limit, one has $u^\infty(p) = \bigl([f(p)],[\infty]\bigr)$, and the energy of $u^\infty$ is essentially the degree of $f$, while the energy of $u_n$ is essentially the degree of $f$ plus the degree of $g$. The number of 'bubbles' is the number of points in the zero divisor of $g$, and this can be arbitrarily large.

  • $\begingroup$ I wonder if it reasonable to think that there is no such example considering a generic target, or at least that the blow-up phenomena are much constrained? $\endgroup$ – Paul Nov 7 '17 at 10:37
  • 2
    $\begingroup$ @Paul: If there is one non-constant harmonic map $f:S^2\to N^n$, where $N^n$ is an arbitrary target, then you can find such examples in the form $f\circ u_n:\Sigma\to N^n$ where $u_n:\Sigma\to S^2$ is a sequence of holomorphic maps of a Riemann surface $\Sigma$ to $S^2$, so blow-up phenomena can never be completely avoided. $\endgroup$ – Robert Bryant Nov 7 '17 at 12:11
  • $\begingroup$ Thanks for the answer, indeed we can loose compactness using the parametrization, but my idea is more about a bad behaviour in the target, is there is any way to glue two harmonic maps for instance, or "equivalently" to find a sequence whose limit consists for instant of an harmonic map and an harmonic sphere in a generic manifold. My guess is that if it is true the junction point, degenerating neck, should be special, for instance a critical point of some curvature? does this question make sense? $\endgroup$ – Paul Nov 7 '17 at 20:11

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