Generalized neck analysis of harmonic maps from surfaces

Abstract

In this paper, we study the behavior of a sequence of harmonic maps from surfaces with uniformly bounded energy on the generalized neck domain. The generalized neck domain is a union of ghost bubbles and annular neck domains, which connects non-trivial bubbles. An upper bound of the energy density is proved and we use it to study the limit of the nullity and index of the sequence.

Appendix A. Some properties of an ODE solution

In this appendix, we show some elementary properties of the solution g(t) to the ordinary differential equation with boundary values

$$\begin{aligned} g''(t)=\gamma ^2 g(t),\qquad g(0)=a \quad \text {and} \quad g(T)=b. \end{aligned}$$

We assume that $\gamma >1/2$ and $T>5$. They are not essential and we assume these for simplicity.

Lemma A.1

There is a universal constant ${\tilde{c}}$ such that for any positive constants a and b with $b\ge a$, we have

$$\begin{aligned} \sup _{[1,T]}\left| (\log g)'\right| \le {\tilde{c}} \end{aligned}$$

(67)

and

$$\begin{aligned} a\le {\tilde{c}} \inf _{[0,1]} g. \end{aligned}$$

(68)

Remark A.2

In general, since b may be very large, even a lot larger than $e^{\gamma T}a$, we can not expect an upper bound of $(\log g)'$ over [0, T]. The observation is that such an upper bound holds for [1, T] regardless of the size of a, b and T.

The proof follows from explicit computation, since we have the following formula for the solution

$$\begin{aligned} g(t)=\frac{a e^{2\gamma T}- be^{\gamma T}}{e^{2\gamma T}-1} e^{-\gamma t} + \frac{b e^{\gamma T}-a}{e^{2\gamma T}-1} e^{\gamma t}. \end{aligned}$$

(69)

Direct computation shows

$$\begin{aligned} g'(t)=- \gamma \frac{a e^{2\gamma T}- be^{\gamma T}}{e^{2\gamma T}-1} e^{-\gamma t} + \gamma \frac{b e^{\gamma T}-a}{e^{2\gamma T}-1} e^{\gamma t} \end{aligned}$$

(70)

and

$$\begin{aligned} \frac{g'(t)}{g(t)}= \gamma \frac{(be^{\gamma T}-a) e^{\gamma t} - (a e^{2\gamma T}-b e^{\gamma T}) e^{-\gamma t}}{(be^{\gamma T}-a) e^{\gamma t} + (a e^{2\gamma T}-b e^{\gamma T}) e^{-\gamma t}}. \end{aligned}$$

(71)

Taking one more derivative, we get

$$\begin{aligned} \left( \frac{g'}{g} \right) ' = \gamma ^2 \frac{4 (be^{\gamma T}-a)(a e^{2\gamma T}-b e^{\gamma T})}{((be^{\gamma T}-a) e^{\gamma t} + (a e^{2\gamma T}-b e^{\gamma T}) e^{-\gamma t})^2}. \end{aligned}$$

(72)

• (i) If a and b are comparable in the sense that $a\le b\le e^{\gamma T}a$, the lemma holds with ${\tilde{c}}=\gamma$. To see this, we notice that in this case

  $$\begin{aligned} {a e^{2\gamma T}- be^{\gamma T}}, {b e^{\gamma T}-a}\ge 0. \end{aligned}$$

  Hence, by (71), we have $\left| (\log g)'\right| \le \gamma$ for all $t\in [0,T]$, from which both (67) and (68) follow.

• (ii) If $b\ge e^{\gamma T}a$, (70) implies that $g'\ge 0$. Hence, g is increasing and (68) follows.

Moreover, (72) implies that $(\log g)''\le 0$. Together with the observation

$$\begin{aligned} (\log g)' (T) = \gamma \frac{b(e^{2\gamma T}+1)-2ae^{\gamma T}}{b(e^{2\gamma T}-1)} \in (0,2\gamma ), \end{aligned}$$

it suffices to bound $(\log g)'(1)$, which we compute

$$\begin{aligned} (\log g)'(1)= & {} \gamma \frac{(b e^{\gamma T}-a) e^\gamma - (a e^{2\gamma T}-b e^{\gamma T})e^{-\gamma }}{(b e^{\gamma T}-a) e^\gamma + (a e^{2\gamma T}-b e^{\gamma T})e^{-\gamma }} \\= & {} \gamma \frac{b e^{\gamma (T+1)} + b e^{\gamma (T-1)} - a e^\gamma - ae^{\gamma (2T-1)}}{b e^{\gamma (T+1)} - b e^{\gamma (T-1)} - a e^\gamma + a e^{\gamma (2T-1)}}\\\le & {} \gamma \frac{b e^{\gamma (T+1)} + b e^{\gamma (T-1)}}{b e^{\gamma (T+1)} - b e^{\gamma (T-1)} - a e^\gamma }. \end{aligned}$$

Finally, we notice that

$$\begin{aligned} a e^\gamma \le b e^{-\gamma T + \gamma }, \end{aligned}$$

which implies that

$$\begin{aligned} (\log g)'(1) \le \frac{2 \gamma }{1- e^{-2\gamma }- e^{-2\gamma T}} \le 4\gamma . \end{aligned}$$

This concludes the proof of the lemma.