Second Inner Variations of Energy and Index of Codimension $2$ Minimal Submanifolds

Many of the results connecting the Allen–Cahn equation to minimal hypersurfaces are now being extended to the codimension $2$ case. In this context, the Abelian Yang–Mills–Higgs equations (sometimes referred to as “$U(1)$-Yang–Mills–Higgs”) are

	$\displaystyle\epsilon^{2}\nabla^{*}\nabla u$	$\displaystyle=\frac{1}{2}(1-|u|^{2})u$		(3)
	$\displaystyle\epsilon^{2}d^{*}F_{\nabla}$	$\displaystyle=i\text{Im}\langle u,\nabla u\rangle_{L}$

and Ginzburg–Landau equations

$$\epsilon^{2}\Delta u=u(|u|^{2}-1)$$

(4)

These equations have been used to approximate minimal submanifolds $Y^{n-2}$. In the Abelian Yang–Mills–Higgs setting, Jaffe and Taubes [15] initially constructed solutions to equation (3) when $M=\mathbb{R}^{2}$, concentrating at a set of points $\{z_{i}\}_{i=1}^{k}$ as $\epsilon\to 0$. In all dimensions, Badran–Del-Pino [1] [2] have completed a gluing construction, paralleling the result of Pacard–Ritore. Parise–Pigati–Stern [21] have established $\Gamma$-convergence properties and Pigati–Stern [22] established weak convergence of the energy measures to stationary integral rectifiable $(n-2)$-varifolds:

In the Ginzburg–Landau setting, Lin–Riviere [18], Betheul–Brezis–Orlandi [3], and Betheul–Orlandi–Smets [4] showed that solutions concentrate about a collection of codimension $2$ minimal submanifolds. This was later refined in the Riemannian setting by Cheng [5], Stern [25] and Pigati–Stern [23]. These authors show that the energy of a sequence of solutions converge to a stationary, but potentially not integral, $n-2$ varifold, plus a diffuse measure on all of $M$.

While the diffuse measure and lack of integrality is differs from the Abelian Yang–Mills–Higgs setting, Cheng [7] and Stern [25] have shown that there exists solutions, $\{u_{\epsilon_{k}}\}$, for which $f=0$ and $V$ is integral. When the minimal submanifold is prescribed, De-Phillipis–Pigati [10] used variational methods to prove the existence of solutions accumulating along codimension $1$ and $2$ in each of the Allen-Cahn, Abelian Yang–Mills–Higgs, and Ginzburg–Landau settings.

Given the parallels between the Allen-Cahn, Abelian Yang–Mills–Higgs, and Ginzburg–Landau equations in $\Gamma$-convergence and convergence of the energy measures, one expects to replicate results which pass geometric information between a sequence of solutions, $\{(u_{\epsilon},\nabla_{\epsilon})$ ( $(\{u_{\epsilon}\}$), and the limiting stationary varifold, $V$. This is the goal of this paper: to recreate the Morse-Index bound of Le and Gaspar in the context of the Ginzburg–Landau and Abelian Yang–Mills–Higgs equations on Riemannian manifolds.

Regularity aside, theorem 1.2 says that equation (5) is typical. By contrast, theorem 1.3 says that equation (6) is atypical, but Cheng [7] and Stern [25] show that $b_{1}(M)=0$ implies $f=0$ in theorem 1.3. In any case, we assume both convergence assumptions to establish morse index bounds. We state our main theorems here:

Remark  We note that Le ([17], theorem 1.5) and Cheng ([6], Proposition 2.6) have done this computation for a sequence of Ginzburg–Landau solutions in the Euclidean and Riemannian setting, respectively. Both authors compute this to investigate stability of Ginzburg-Landau solutions. We also note that Cheng ([6], Proposition 5.2) computes the second inner variation for Abelian Yang-Mills-Higgs in a different form, again to prove stability. We include our own derivations since they emphasize the parallels between the Abelian Yang–Mills–Higgs and Ginzburg–Landau cases, unified by the convergence of the stress energy tensor.

For any $U\subseteq M$ with $U\cap\text{reg}(V)\neq\emptyset$, let $\{\lambda_{p}(U)\}$ denote the eigenvalues of the Jacobi operator of $V$ acting on normal vector fields supported on $U\cap\text{reg}(V)$. Similarly, for a sequence of critical points $\{(u_{\epsilon_{k}},\nabla_{\epsilon_{k}})\}$ (resp. $\{u_{\epsilon_{k}}\}$ ) to the Abelian Yang–Mills–Higgs (resp. Ginzburg–Landau) equations, let $\lambda_{AYMH,p}^{\epsilon_{k}}(U)$ (resp. $\lambda_{GL,p}^{\epsilon_{k}}$) denote the eigenvalues of the linearized Abelian Yang–Mill–Higgs (resp. Ginzburg–Landau) operator at $(u_{\epsilon_{k}},\nabla_{\epsilon_{k}})\Big{|}_{U}$ (resp. $u_{\epsilon_{k}}\Big{|}_{U}$).

We can immediately apply this to any of the solutions constructed in [10], [7], [25], [2] to give a lower bound on the morse index of the constructed solutions, $\{u_{\epsilon}\}$ or $\{(u_{\epsilon},\nabla_{\epsilon})\}$, in terms of the morse index of the limiting varifold.

The techniques to prove theorems 2.1 2.2 are almost identical to that of Gaspar [11] and Hiesmayr [13], expect we replace the Allen-Cahn equation with the Abelian Yang–Mills–Higgs and Ginzburg–Landau equations. We also substitute proposition 2.2 of Gaspar [11] with knowledge of the limit of the stress-energy tensor for the Abelian Yang–Mills-Higgs and Ginzburg–Landau equations. We also remark that up to the appropriate constant, the integral term in (7) simplifies to that of Gaspar’s ([11], Proposition 3.3) and Le’s ([16], Theorem 1) when the normal bundle is $1$-dimensional and $X$ lies in the normal bundle.

In the Ginzburg–Landau case, we compute using (9):

	$\displaystyle\frac{d^{2}}{dt^{2}}\left(|\nabla u^{t}|^{2}\circ\Phi^{t}\right)(x)\Big{|}_{t=0}$	$\displaystyle=\ddot{g}^{ij}u_{i}u_{j}\Big{|}_{x}$
		$\displaystyle=\langle 2(\dot{g}\circ\dot{g})-\ddot{g},\nabla u^{*}\nabla u\rangle$

here,

$$(\dot{g}\circ\dot{g})_{ij}:=\dot{g}_{ik}g^{k\ell}\dot{g}_{\ell j}$$

So that

	$\displaystyle\frac{d^{2}}{dt^{2}}E_{\text{GL},\epsilon}(u^{t}_{\epsilon})\Big{|}_{t=0}$	$\displaystyle=\int_{M}\frac{1}{2}\frac{d^{2}}{dt^{2}}\left(|\nabla^{t}u^{t}|^{2}\circ\Phi^{t}\right)dVol+\frac{d}{dt}\left(|\nabla^{t}u^{t}|^{2}\circ\Phi^{t}\right)\Big{|}_{t=0}\cdot\frac{d}{dt}(\Phi^{t})^{*}(dVol_{g})\Big{|}_{t=0}+e_{\epsilon}(u)\frac{d^{2}}{dt^{2}}(\Phi^{t})^{*}(dVol_{g})\Big{|}_{t=0}$
		$\displaystyle=\int_{M}\frac{1}{2}\langle 2(\dot{g}\circ\dot{g})-\ddot{g},S_{GL,\epsilon}\rangle dVol_{M}$
		$\displaystyle+\int_{M}-\langle\dot{g},S_{GL,\epsilon}\rangle\text{div}(X)dVol$
		$\displaystyle+\int_{M}e_{\text{GL},\epsilon}(u_{\epsilon})\left[\text{div}(\nabla_{X}X)-\text{Ric}(X,X)-T_{TM}(X,X)+\text{div}(X)^{2}\right]dVol$

In the Abelian Yang–Mills–Higgs case, we proceed analogously:

	$\displaystyle\frac{d^{2}}{dt^{2}}E_{\text{AYMH},\epsilon}(u,A)\Big{|}_{t=0}$	$\displaystyle=\int_{M}\frac{1}{2}\langle 2(\dot{g}\circ\dot{g})-\ddot{g},S_{AYMH,\epsilon}\rangle dVol_{M}$
		$\displaystyle+\int_{M}-\frac{1}{2}\langle\dot{g},S_{AYMH,\epsilon}\rangle\text{div}(X)dVol$
		$\displaystyle+\int_{M}e_{\text{AYMH},\epsilon}(u_{\epsilon})\left[\text{div}(\nabla_{X}X)-\text{Ric}(X,X)-T_{TM}(X,X)+\text{div}(X)^{2}\right]$

We now reduce the second inner variation, using our lemmas about the stress-energy tensor.

Proof of theorem 2.1:
From lemma 2 and equation (6), we have that

	$\displaystyle\lim_{\epsilon\to 0}\frac{d^{2}}{dt^{2}}E_{\text{GL},\epsilon}(u^{t})\Big{|}_{t=0}$	$\displaystyle=\pi\sum_{i=1}^{q}m_{i}\int_{\Gamma_{i}}\Big{(}\frac{1}{2}\left[2\text{tr}_{N\Gamma_{i}}(\dot{g}\circ\dot{g})-\text{tr}_{N\Gamma_{i}}(\ddot{g})\right]-\text{tr}_{N\Gamma_{i}}(\dot{g})\text{div}(X)$
		$\displaystyle\qquad+[\text{div}(\nabla_{X}X)-\text{Ric}(X,X)-T_{TM}(X,X)+\text{div}_{TM}(X)^{2}]\Big{)}dVol$

For $P,W\subseteq TM$ subbundles and $X\in\Gamma(TM)$, let

$$||\nabla_{P}^{W}X||^{2}:=\text{tr}_{P}\langle\Pi^{W}\nabla_{\cdot}X,\Pi^{W}\nabla_{\cdot}X\rangle$$

At a point $x\in M$, if $\{v_{i}\}_{i=1}^{k}$ is an orthonormal basis for $P_{x}$ and $\{w_{j}\}_{j=1}^{\ell}$ orthonormal for $W_{x}$, the above becomes

	$\displaystyle||\nabla_{P}^{W}X||^{2}\Big{|}_{x}$	$\displaystyle=\sum_{i=1}^{d}\langle\Pi^{W}\nabla_{v_{i}}X,\Pi^{W}\nabla_{v_{i}}X\rangle\Big{|}_{x}$
		$\displaystyle=\sum_{i=1}^{d}\sum_{j=1}^{\ell}\langle\nabla_{v_{i}}X,w_{j}\rangle^{2}\Big{|}_{x}$

We compute in an orthonormal basis:

	$\displaystyle\text{tr}_{N\Gamma_{i}}(\dot{g}\circ\dot{g})$	$\displaystyle=\sum_{i=n-1}^{n}(\dot{g}\circ\dot{g})_{ii}$
		$\displaystyle=\sum_{i=n-1}^{n}\sum_{j=1}^{n}\dot{g}_{ij}^{2}$
		$\displaystyle=\sum_{i=n-1}^{n}\sum_{j=1}^{n}[\langle\nabla_{i}X,\partial_{j}\rangle^{2}+\langle\nabla_{j}X,\partial_{i}\rangle^{2}+2\langle\nabla_{i}X,\partial_{j}\rangle\langle\nabla_{j}X,\partial_{i}\rangle]$
		$\displaystyle=||\nabla_{N\Gamma_{i}}^{TM}X||^{2}+||\nabla_{TM}^{N\Gamma_{i}}X||^{2}+2\text{tr}_{N\Gamma_{i}}\langle\nabla_{\nabla_{\cdot}X}X,\cdot\rangle$
	$\displaystyle\text{tr}_{N\Gamma_{i}}(\ddot{g})$	$\displaystyle=\sum_{i=n-1}^{n}2\langle\nabla_{i}\nabla_{X}X,\partial_{i}\rangle+2\langle\nabla_{i}X,\nabla_{i}X\rangle-2R(X,\partial_{i},X,\partial_{i})$
		$\displaystyle=2\text{div}_{N\Gamma_{i}}(\nabla_{X}X)+2||\nabla_{N\Gamma_{i}}^{TM}X||^{2}-2\text{tr}_{N\Gamma_{i}}R(X,\cdot,X,\cdot)$
	$\displaystyle\text{tr}_{N\Gamma_{i}}(\dot{g})$	$\displaystyle=2\sum_{i=n-1}^{n}\langle\nabla_{i}X,\partial_{i}\rangle$
		$\displaystyle=2\text{div}_{N\Gamma_{i}}(X)$

In sum

	$\displaystyle\lim_{\epsilon\to 0}\frac{d^{2}}{dt^{2}}E_{\text{GL},\epsilon}(u^{t})\Big{|}_{t=0}$	$\displaystyle=\pi\sum_{i=1}^{q}m_{i}\int_{\Gamma_{i}}\Big{(}\left[||\nabla_{TM}^{N\Gamma_{i}}X||^{2}+2\text{tr}_{N\Gamma_{i}}\langle\nabla_{\nabla_{\cdot}X}X,\cdot\rangle-\text{div}_{N\Gamma_{i}}(\nabla_{X}X)+\text{tr}_{N\Gamma_{i}}R(X,\cdot,X,\cdot)\right]$
		$\displaystyle\qquad\quad-2\text{div}_{N\Gamma_{i}}(X)\text{div}(X)$
		$\displaystyle\qquad\quad+[\text{div}(\nabla_{X}X)-\text{Ric}(X,X)-T_{TM}(X,X)+\text{div}_{TM}(X)^{2}]\Big{)}dVol_{\Gamma_{i}}$

With some cancellation, we group

	$\displaystyle\text{tr}_{N\Gamma_{i}}R(X,\cdot,X,\cdot)-\text{Ric}(X,X)$	$\displaystyle=-\text{tr}_{T\Gamma_{i}}R(X,\cdot,X,\cdot)$
	$\displaystyle\text{div}_{TM}(X)^{2}-2\text{div}_{N\Gamma_{i}}(X)\text{div}(X)$	$\displaystyle=\text{div}_{T\Gamma_{i}}(X)^{2}-\text{div}_{N\Gamma_{i}}(X)^{2}$
	$\displaystyle\text{div}(\nabla_{X}X)-\text{div}_{N\Gamma_{i}}(\nabla_{X}X)$	$\displaystyle=\text{div}_{T\Gamma_{i}}(\nabla_{X}X)$
	$\displaystyle||\nabla_{TM}^{N\Gamma_{i}}X||^{2}$	$\displaystyle=||\nabla_{T\Gamma_{i}}^{N\Gamma_{i}}X||^{2}+||\nabla_{N\Gamma_{i}}^{N\Gamma_{i}}X||^{2}$
	$\displaystyle||\nabla_{N\Gamma_{i}}^{N\Gamma_{i}}X||^{2}+2\text{tr}_{N\Gamma_{i}}\langle\nabla_{\nabla_{\cdot}X}X,\cdot\rangle-T_{TM}(X,X)$	$\displaystyle=-T_{T\Gamma_{i}}(X,X)+\frac{1}{2}||\dot{g}||_{N\Gamma_{i}}^{2}$

For $T$ as in equation (8). This gives

	$\displaystyle\lim_{\epsilon\to 0}\frac{d^{2}}{dt^{2}}E_{\text{GL},\epsilon}(u^{t})\Big{|}_{t=0}$	$\displaystyle=\pi\sum_{i=1}^{q}m_{i}\int_{\Gamma_{i}}\Big{(}-\text{tr}_{T\Gamma_{i}}R(X,\cdot,X,\cdot)+\text{div}_{T\Gamma_{i}}(X)^{2}+\text{div}_{T\Gamma_{i}}(\nabla_{X}X)+||\nabla_{T\Gamma_{i}}^{N\Gamma_{i}}X||^{2}-T_{T\Gamma_{i}}(X,X)\Big{)}$
		$\displaystyle\quad+\Big{(}-\text{div}_{N\Gamma_{i}}(X)^{2}+\frac{1}{2}||\dot{g}||_{N\Gamma_{i}}^{2}\Big{)}$
		$\displaystyle=\pi\sum_{i}m_{i}\left[D^{2}A\Big{|}_{\Gamma_{i}}(X,X)+\int_{\Gamma_{i}}\Big{(}-\text{div}_{N\Gamma_{i}}(X)^{2}+\frac{1}{2}||\dot{g}||_{N\Gamma_{i}}^{2}\Big{)}dVol\right]$

having used equation (10). Note that the error term is non-negative: let $\{\partial_{1},\partial_{2}\}$ a basis for $N\Gamma_{i}$ which is orthonormal when restricted to $\Gamma_{i}$. We have:

	$\displaystyle-\text{div}_{N\Gamma_{i}}(X)^{2}+\frac{1}{2}||\dot{g}||_{N\Gamma_{i}}^{2}$	$\displaystyle=\sum_{i,j=n-1}^{n}-\langle\nabla_{i}X,\partial_{i}\rangle\langle\nabla_{j}X,\partial_{j}\rangle+\frac{1}{2}[\langle\nabla_{i}X,\partial_{j}\rangle+\langle\nabla_{j}X,\partial_{i}\rangle]^{2}$
		$\displaystyle=[\langle\nabla_{n-1}X,\partial_{n-1}\rangle-\langle\nabla_{n}X,\partial_{n}\rangle]^{2}+[\langle\nabla_{n-1}X,\partial_{n}\rangle+\langle\nabla_{n}X,\partial_{n-1}\rangle]^{2}$
		$\displaystyle=\frac{1}{4}[\dot{g}_{n-1,n-1}-\dot{g}_{n,n}]^{2}+\dot{g}_{n-1,n}^{2}$
		$\displaystyle=[\langle\nabla_{n-1}X,\partial_{n-1}\rangle-\langle\nabla_{n}X,\partial_{n}\rangle]^{2}+[\langle\nabla_{n-1}X,\partial_{n}\rangle+\langle\nabla_{n}X,\partial_{n-1}\rangle]^{2}$	$\displaystyle\geq 0$

We note that the error term vanishes when $\nabla_{i}X\Big{|}_{\gamma}=0$, e.g. if $X\Big{|}_{\Gamma_{i}}=f^{1}(s)\partial_{1}+f^{2}(s)\partial_{2}$, where $s$ is a coordinate on $\Gamma_{i}$, and $\partial_{1},\partial_{2}$ are normal vector fields corresponding to a parallel frame on $\Gamma_{i}$. Such a vector field can be extended to a vector field on $M$ via a bump function in a tubular neighborhood of $\Gamma_{i}$ (see [11], Appendix). We also note that this is the same error term in Cheng [6], Proposition 2.6, as well as Le [17], Theorem 1.5.

The reduction of $\lim_{\epsilon\to 0}\frac{d^{2}}{dt^{2}}E_{AYMH,\epsilon}(u^{t},\nabla^{t})$ is identical with the same result (but a normalizing factor of $2\pi$) since the reduction only depends on the limit of the energy measure, the stress energy tensor, and derivatives of $g^{t}$.