Variation Formulae for Perimeter and Volume Densities

For Rn+1 with volume density f and perimeter density g, for a normal variation u of a surface with classical mean curvature H, the first variation of volume and perimeter are given by:

\delta ^1V=-\int uf,

\delta ^1P=-\int [(g/f)nH - (1/f)(\partial g/\partial n)] uf.

For a volume-preserving variation, the second variation of perimeter is given by:
\delta ^2P=\int g|\nabla u|^2-g|\sigma|^2u^2-f\frac{\partial (g/f)}{\partial n}u^2nH+u^2\frac{\partial ^2g}{\partial n^2}-\frac 1fu^2\frac{\partial f}{\partial n}\frac{\partial g}{\partial n},

where \sigma is the second fundamental form, so that \sigma^2 is the sum of the squares of the principal curvatures.

The proof is the same as for the case of simple density, f=g; see Rosales et al. [R], Section 3.

It follows that balls about the origin are stable for a radial density if and only if:

a simple density (f(r)=g(r)) is log convex;

a perimeter density g(r) (f=1) satisfies (n/r)g‘ + g” ≥ 0;

a volume density f(r) (g=1) is nonincreasing;

a conformal change of metric f(r) = λn+1, g(r) = λn satisfies −λλ’/r + λλ” − 2λ’2 ≥ 0.

The conformal case can be checked by using the standard second variation formula for a Riemannian metric:

\delta ^2P=\int |\nabla u|^2-|\sigma|^2u^2-Ric(u,u).

In the planar case (n=1), the curvature κ is given by

\kappa = -\frac{dP}{dA} = -\frac{\lambda + r\lambda^\prime}{\lambda ^2r}

and the Gauss curvature is given by the formula

G=-\lambda^{-2}\Delta log\lambda
= λ-4(λλ’/r + λλ” −2λ’2).

Therefore |σ|2 = nκ2 and Ric(u,u) = nGu2. Also, the gradient of u is larger by a factor of λ in the background metric. Now the second variation formula yields the same condition for stability of spheres. (For the lowest eigenvalue, the gradient term and the 1/r2 term cancel out.)

Remark February 14, 2011. In the plane with density rp (p>0), isoperimetric regions are balls with the origin on the boundary [Dahlberg et al.]. For unequal radial volume and perimeter densities, such balls are not even in equilibrium.

Corollary June 25, 2016. In a cone over a round n-sphere of mean curvature 1/a, volume density f(r), and perimeter density g(r),  a geodesic sphere about the origin of radius r has nonnegative second variation if and only if

(1)  n(a^2 -1) + nr(f/g)(g/f)' + r^2 g''/g - r^2 (1/fg)(f'g') \ge 0.

For a single density g = f, (1) becomes

(2) n(a^2 - 1)+r^2 (log f)'' \ge 0.

which for a=1 reduces to the standard log convexity condition. For volume density f = r^m and perimeter density g = r^k, (1) and (2) become

(1a)  m \le k-1+na^2 /(k+n),

in agreement with Alvino et al. [A, (5.53)] when a=1;

(2a) n(a^2-1) \ge m.

In the conformal case m=k(n+1)/n, (1a) becomes k \le n(a-1). For n=1, the cone is equivalent to the \theta = \pi /a sector, and (2a) becomes \theta \le \pi /\sqrt{m+1} of Diaz et al. [D, Prop. 4.16].

Proof. The lowest mode on S^n comes from translation in R^{n+1}, i.e. u = x_1, with |\nabla u|^2 on the average n times as large as u^2 when a=1 and n a^2 times as large in general. |\sigma|^2 is n/r^2.

Theorem [E, 6.3]. Let S be a smooth Riemannian disk, sphere, or annulus of revolution with metric ds^2 = dr^2 + \phi(r)^2 d\theta^2 and density f(r). Then the circle of revolution at distance r has nonnegative second variation if and only if

Q(r) = \phi'(r)^2 - \phi(r)\phi''(r) - \phi(r)^2 (log f)'' \le 1.

Proposition (May 7, 2019). Similarly, in an (n+1)D ambient of revolution, the sphere at distance r has nonnegative second variation if and only if

Q(r) = \phi'(r)^2 - \phi(r)\phi''(r) - \phi(r)^2 (log f)''/n \le 1.

Proof. The Gauss curvature of a radial section is -\phi''/\phi [E, 6.3], so the outward Ricci curvature is -n\phi''/h. Each principal curvature equals the mean curvature

dA/ndV = (log A)'/n = log(\phi^n)'/n = \phi'/\phi,

so the second fundamental form II satisfies II^2 = nh'^2/h^2. By Rosales et al. [R, Rmk. 3.7], the second variation of a sphere S about the origin for a normal (i.e. radial) variation u that preserves volume to first order (∫u = 0) satisfies

\delta^2(u) = \int |\nabla u|^2 - u^2(Ric(N,N) - (log f)'' + II^2).

By the generalized Wirtinger Inequality [O, (3.1)] (Poincaré Inequality), the second variation is nonnegative if and only if

n/h^2 \le Ric(N,N) - (log f)'' + II^2 = -n\phi''/\phi - (log f)'' + n\phi'^2/\phi^2,

i.e.,

1 \ge \phi'^2 - \phi\phi'' - \phi^2(log f)''/n. QED.

Corollary. In an (n+1)D ambient of revolution, for decreasing ambient curvature and convex density, spheres about the origin are stable for fixed volume, while for increasing curvature and concave density, they are unstable.

Proof. For infinitesimal spheres, the ambient is Euclidean and Q = 0. For constant density f, Q and the Gauss curvature K of a radial section satisfy Q' = \phi^2 K' (trivial one-line computation as in [R, Lemma 1.6]), [E, Sect. 6], [M]). In particular, Q and K increase or decrease together. The convexity or concavity of the density only causes Q to be still smaller or larger.

[A] A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M.R. Posteraro, Some isoperimetric inequalities on R^N with respect to weights |x|^\alpha, arXiv.org (2016). .

[D] Alexander Díaz, Nate Harman, Sean Howe, David Thompson. Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589–619; arXiv.org (2010).

[E] Max Engelstein, Anthony Marcuccio, Quinn Maurmann, and Taryn Pritchard, Isoperimetric problems on the sphere and on surfaces with density, New York J. Math. 15 (2009), 97–123.

[M] Frank Morgan,  Isoperimetric symmetry breaking: a counterexample to a generalized form of the log-convex density conjecture, Anal. Geom. Metr. Spaces 4 (2016), 314-316.

[O] Robert Osserman, The Isoperimetric Inequality, Bull. Amer. Math. Soc. 84 (1978), 1182-1238.

[R] César Rosales, Antonio Cañete, Vincent Bayle, and Frank Morgan. On the isoperimetric problem in Euclidean space with density. Calc. Var. PDE 31 (2008), 27-46; arXiv.org (2006).

One Comment

  1. The Log-Convex Density Conjecture at Frank Morgan:

    […] When rephrased to state that all balls about the origin are isoperimetric if stable, the conjecture naturally generalizes to perimeter density, volume density, or both (different densities on volume and perimeter). Examples are provided by [Betta et al.] and [Díaz et al. Prop. 4.24]. Actually [June 22, 2010], stability of balls about the origin is not quite enough; you need to require smoothness or at least log-convexity of the density at the origin. For example, in the plane with density r-1, isoperimetric regions do not exist, and in the plane with density exp(r2-2r+2), isoperimetric regions for small volume are approximately round balls about the minimum density at r = 1. For perimeter density g(r) on Rn+1, the stability condition for balls about the origin is (n/r)g‘ + g” ≥ 0 for r>0; one should further assume that the inequality holds in some sense at 0. The situation is simpler for area density, where the stability condition (area density nonincreasing) trivially implies that balls about the origin are isoperimetric. (These stability conditions follow from the second variation formulae.) […]

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