On the variation of curvature functionals in a space form with application to a generalized Willmore energy

Abstract

Functionals involving surface curvature are important across a range of scientific disciplines, and their extrema are representative of physically meaningful objects such as atomic lattices and biomembranes. Inspired in particular by the relationship of the Willmore energy to lipid bilayers, we consider a general functional depending on a surface and a symmetric combination of its principal curvatures, and provided the surface is immersed in a 3-D space form of constant sectional curvature. We calculate the first and second variations of this functional, extending known results and providing computationally accessible expressions given entirely in terms of the basic geometric information found in the surface fundamental forms. Further, we motivate and introduce the p-Willmore energy functional, applying the stability criteria afforded by our calculations to prove a result about the p-Willmore energy of spheres.

Appendix

Note the following conventions:

• Einstein summation is assumed throughout, so that any index repeated twice in an expression (once up and once down) will be contracted over its appropriate range.

• Differentiation of a function f with respect to the variable $x^j$ is denoted by $f_j$.

• Given a t-parametrized variation, the variational derivative operator is denoted by $\delta = (\mathrm{d}/dt)\big |_{t=0}$.

Proof

(Proof of Lemma 1) First, there is the variation of the metric: without loss of generality, assume $\langle \mathbf {r}_i,\mathbf {r}_j \rangle = 0$ on $M_t$. Using $\langle \mathbf {N}, \mathbf {r}_j \rangle = 0$,

$$\begin{aligned} \begin{aligned} \delta g_{ij}&= \frac{d}{\mathrm{d}t} \big \langle \mathbf {r}_i, \mathbf {r}_j \big \rangle \bigg |_{t=0} = 2 \big \langle (\delta \mathbf {r})_i, \mathbf {r}_j \big \rangle = 2 \langle u_i \mathbf {N} + u\mathbf {N}_i , \mathbf {r}_j \rangle \\&= 2u_i \langle \mathbf {N}, \mathbf {r}_j \rangle + 2u \langle \mathbf {N}_i, \mathbf {r}_j \rangle = -2uh_{ij}. \end{aligned} \end{aligned}$$

(62)

Since $g^{il}g_{lk}=\delta _{k}^i$, it follows that

$$\begin{aligned} (\delta g^{il})g_{lk} = -g^{il}(\delta g_{lk}) = 2ug^{il}h_{lk}, \end{aligned}$$

(63)

and hence

$$\begin{aligned} \delta g^{ij} = 2ug^{il}g^{jk}h_{lk} = 2uh^{ij}. \end{aligned}$$

(64)

Using this, there is the variation of the area element: Recall the Jacobi formula

$$\begin{aligned} \begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \det A = \det A \; \mathrm {tr} \left( A^{-1} \frac{\mathrm{d}A}{\mathrm{d}t}\right) . \end{aligned} \end{aligned}$$

(65)

Letting $\mathbf {g} = g_i^j$ be the matrix representation of the metric, it follows that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\det (\mathbf {g}) = \det (\mathbf {g}) \; \mathrm {tr} \left( \mathbf {g}^{-1} \frac{d\mathbf {g}}{\mathrm{d}t}\right) = \det (\mathbf {g}) (-2ug^{ij}h_{ij}) = -4 Hu \det (\mathbf {g}). \end{aligned}$$

(66)

Using (66), the variation of the surface area functional $\mathcal {A}$ is seen to be

$$\begin{aligned} \begin{aligned} \delta \mathcal {A}&= \delta \int _M \mathrm{d}S = \int _U \delta \sqrt{\det ({\mathbf {g}})} \, dA = \int _U \frac{1}{2\sqrt{\det ({\mathbf {g}})}} \, \delta (\det ({\mathbf {g}})) \, dA \\&= \int _U 2 Hu \sqrt{\det ({\mathbf {g}})} \, dA = \int _M -2Hu \, \mathrm{d}S. \end{aligned} \end{aligned}$$

(67)

Using (67) and observing the commutativity of d and $\delta$ yields the variation of the area element $\mathrm{d}S$,

$$\begin{aligned} \delta (\mathrm{d}S) = d(\delta \mathcal {A}) = d \int _M -2 Hu \, \mathrm{d}S = -2 Hu \, \mathrm{d}S. \end{aligned}$$

(68)

It is now necessary to compute the variation of the shape operator $h = h_{ij}\, dx^i \otimes dx^j$. Observe,

$$\begin{aligned} \delta (h_{ij}) = \delta \langle \mathbf {N}, \mathbf {r}_{ij} \rangle = \langle \delta \mathbf {N}, \mathbf {r}_{ij} \rangle + \langle \mathbf {N}, \delta \mathbf {r}_{ij} \rangle . \end{aligned}$$

(69)

It is advantageous to compute each term of (69) separately. Since $\mathbf {r}_i,\mathbf {r}_j$ is a basis for TM at each point, the variation of the normal field can be expressed as $\delta \mathbf {N} = c^i\mathbf {r}_i$ for some functions $c^i$, so that

$$\begin{aligned} \langle \delta \mathbf {N}, \mathbf {r}_j \rangle = \langle c^i \mathbf {r}_i, \mathbf {r}_j \rangle = c^i = -\langle \mathbf {N}, \delta \mathbf {r}_j \rangle = -\langle \mathbf {N}, u_i \mathbf {N} + u\mathbf {N}_i \rangle = -u_i, \end{aligned}$$

(70)

where it was used again that $\langle \mathbf {N}, \mathbf {r}_j \rangle = \langle \mathbf {N}, \mathbf {N}_j \rangle = 0$. It follows that $\delta \mathbf {N} = -g^{ij}u_i\mathbf {r}_j$, whereby using (9) and working in normal coordinates one sees

$$\begin{aligned} \big \langle \delta \mathbf {N}, \mathbf {r}_{ij} \big \rangle = -\big \langle u^l\mathbf {r}_l, (h_{ij}\mathbf {N} + \varGamma _{ij}^k \mathbf {r}_k - g_{ij}k_0\mathbf {r}) \big \rangle = g_{ij}k_0u. \end{aligned}$$

(71)

Further, by (10)

$$\begin{aligned} \begin{aligned} \langle \mathbf {N},\delta \mathbf {r}_{ij} \rangle&= \langle \mathbf {N}, (u\mathbf {N})_{ij} \rangle = u_{ij}+ u \langle \mathbf {N}, \mathbf {N}_{ij} \rangle = u_{ij} -u\langle h_i^l \mathbf {r}_l, h_j^k \mathbf {r}_k \rangle \\&= u_{ij} - uh_{il}h^l_j. \end{aligned} \end{aligned}$$

(72)

Therefore, the variation of the second fundamental form is

$$\begin{aligned} \delta (h_{ij}) = \langle \delta \mathbf {N}, \mathbf {r}_{ij} \rangle + \langle \mathbf {N}, \delta \mathbf {r}_{ij} \rangle = g_{ij}k_0u + u_{ij} - uh_{il}h_j^l, \end{aligned}$$

(73)

and it is now straightforward to compute $\delta (2H)$. Indeed, it follows that

$$\begin{aligned} \begin{aligned} \delta (2H)&= \delta (g^{ij}h_{ij}) = (\delta g^{ij})h_{ij} + g^{ij}(\delta h_{ij}) \\&= 2uh^{ij}h_{ij} + g^{ij}(u_{ij} - uh_{il}h_j^l + g_{ij}k_0u)\\&= \varDelta {u}+u(4H^2-2K+4k_0). \end{aligned} \end{aligned}$$

(74)

Further, there is the variation of the norm of the second fundamental form,

$$\begin{aligned} \begin{aligned} \delta |h|^2&= \delta (h^{ij}h_{ij}) = \delta (g^{ik}g^{jl}h_{kl}h_{ij}) = 2uh^{ik}g^{jl}h_{kl}h_{ij} + 2uh^{jl}g^{ik}h_{kl}h_{ij} \\&\quad + g^{ik}g^{jl}(g_{kl}k_0u + u_{kl} - uh_{ks}h^s_l)h_{ij} + g^{ik}g^{jl}h_{kl}(g_{ij}k_0u + u_{ij} - uh_{is}h^s_j) \\&= 2uh^{ik}h^j_k h_{ij} + 4Huk_0 + 2\langle h,\text {Hess}\,u\rangle \\&= 2(8H^3-6HK+8Hk_0)u + 2\langle h,\text {Hess}\,u \rangle , \end{aligned}\qquad \end{aligned}$$

(75)

where it was used that $8H^3 = (\kappa _1+\kappa _2)^3 = \kappa _1^3 + \kappa _2^3 + 6H(K-k_0)$. The variation of the extrinsic Gauss curvature $K_E$ now follows, since $\delta |h|^2 = \delta (4H^2 - 2K + 2k_0)$, so

$$\begin{aligned} \delta K_E = 2H\,\delta (2H) - \frac{1}{2}\delta |h|^2 = 2H\varDelta u - \langle h,\text {Hess}\,u\rangle + 2HKu. \end{aligned}$$

(76)

Since the variation of the intrinsic Gauss curvature K satisfies $\delta K = \delta (K_E + k_0)$ and $k_0$ is constant, we have $\delta K = \delta K_E$, completing the calculation. $\square$