Frank Morgan

[“Soap Bubbles and Mathematics,” written for the Spanish Math Carnival]

¿Por qué son las pompas de jabón tan perfectamente redondas?

Continue reading ‘Pompas de Jabón y las Matemáticas’ »

An interesting class of densities for the isoperimetric problem on Rⁿ is exp(±r^a). The Log-convex Density Conjecture says that spheres about the origin are isoperimetric if and only if 0≤a≤1 with the minus sign or a ≥ 1 with the plus sign, known for a=2 [Borell; see Rosales et al.] and in R² for a≥2 [Maurmann-Morgan]. Henceforth we’ll focus on R². Continue reading ‘Density exp(±r^a)’ »

Rodrigo Banuelos suggested studying the isoperimetric problem for the radial density 1/(1+r²) corresponding to the square root of the Laplacian just as the most important Gaussian density corresponds to the Laplacian itself.

Proposition. Consider Rⁿ with density 1/(1+r²). For n > 1 minimizers of perimeter for given volume do not exist: the perimeter can go to zero as the region goes off to infinity. On the line, for more than half the volume the minimizer is a ball about the origin, for less than half, the complement, for exactly half, the ball, its complement, or a half-line. In particular, balls about the origin are minimizing while stable, up to radius 1, with (log density)” = 2(x²-1)/(x²+1)². Continue reading ‘Density 1/(1+r^2)’ »

Prof. Alberto Bressan at Penn State offers a $500 prize for proving the optimal firebreak to isolate a forest fire that begins in the unit disc. I guess the answer is the heart shape of his Figure 2, the upper half of which he knows to be optimal in the half-plane. (Happy Valentine’s Day.)

Note added February 7, 2011. The announced counterexample was wrong. It does not satisfy the hypothesis of the conjecture. Geodesic circles about the origin in Enneper’s surface are unstable.

The proposed generalization of the Log-convex Density Conjecture to separate densities for volume and perimeter fell at the centennial congress of the Royal Spanish Mathematical Society.  The counterexample—Enneper’s Surface—emerged from discussions with participants in the Geometric Analysis session, notably Manuel Ritoré, Antonio Ros, Cesar Rosales, and Antonio Cañete.

Another way of stating the hypothesis for a single density for volume and perimeter is that the generalized Ricci curvature, equal to the Riemannian Ricci curvature (0 for Euclidean space) minus the Hessian of the log of the density be nonpositive. General separate perimeter and volume densities are equivalent to a conformal change of metric together with a density. The counterexample, Enneper’s Surface, is conformally the plane (with unit density) and intrinsically a surface of revolution. As a minimal surface, it has negative curvature. Since the curvature is negative at the origin and approaches 0 at infinity, small balls about the origin have larger perimeter than small balls near infinity, and the generalized conjecture fails.

Preliminary announcement February 4, 2011.

Conjecture. The least perimeter to enclose given volume inside an open ball in Rⁿ is greater than inside any other convex body of the same volume. Continue reading ‘Convex Body Isoperimetric Conjecture’ »

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:

,

.

For a volume-preserving variation, the second variation of perimeter is given by:
,

where is the second fundamental form, so that is the sum of the squares of the principal curvatures. Continue reading ‘Variation Formulae for Perimeter and Volume Densities’ »

Here’s a proof of the infinitude of primes that occurred to me when for some reason Delta upgraded me to First Class on a flight April 2. Can anyone provide a reference?

Suppose that the set P of primes and 1 has just n+1 elements. Now every number at most 2^k can be obtained by choosing k elements of P with replacement, which can be done in (k+n choose n) ways. Therefore

2^k ≤ (k+n choose n) ≤ (k+n)ⁿ ,

which fails for k large.

Announcement: proof posted 19 November 2013 by Gregory R. Chambers, student of Regina Rotman and Alex Nabutovsky at Toronto. By spherical symmetrization the problem reduces to planar curves, which he studies very intelligently in great detail. The main idea is that if the generating curve is not a circle about the origin, then from its maximum (on the axis of symmetry) it spirals inward and it eventually turns through 2π before returning to the axis, contradiction. See his 30-second video. “Proof of the Log-Convex Density Conjecture,” J. Eur. Math. Soc. 21 (2019), 2301–2332.

Update August 2, 2022. arXiv:2208.00195. “Approaching the isoperimetric problem in H^m_ℂ via the hyperbolic log-convex density conjecture” by Lauro Silini. Abstract: We prove that geodesic balls centered at some base point are isoperimetric in the real hyperbolic space H^n_ℝ endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the result by G. R. Chambers for log-convex densities on ℝ^n. As an application we prove that in the complex and quaterionic hyperbolic spaces, geodesic balls are isoperimetric in a class of sets enjoying a suitable notion of radial symmetry. The hyperbolic plane case was proved by Igor McGillivray: arXiv:1712.07690, “A weighted isoperimetric inequality on the hyperbolic plane.”

Ken Brakke’s Log-Convex Density Conjecture [Rosales et al. Conj. 3.12] says that in Euclidean space with radial log-convex density f(r), balls about the origin are isoperimetric.

A density is just a positive continuous function used to weight volume and perimeter. Log convexity just means that log f is convex. Balls isoperimetric means that any other region of the same weighted volume has no less weighted perimeter.

Log convexity is necessary because it is equivalent to stability. The question is whether it is sufficient. Continue reading ‘The Log-Convex Density Conjecture’ »