Neumann eigenvalue sums on triangles are (mostly) minimal for equilaterals

TL;DR

This paper proves that for triangles with a fixed diameter, the equilateral triangle generally minimizes the sum of the first few Neumann Laplacian eigenvalues, especially for n ≥ 3, highlighting geometric spectral optimization.

Contribution

It establishes that the equilateral triangle minimizes the sum of the first n Neumann eigenvalues for n ≥ 3 among all triangles of given diameter.

Findings

Equilateral triangle minimizes the sum of the first n eigenvalues for n ≥ 3.

The second eigenvalue's minimum is achieved by a degenerate isosceles triangle.

The third eigenvalue is minimized by the equilateral triangle.

Abstract

We prove that among all triangles of given diameter, the equilateral triangle minimizes the sum of the first $n$ eigenvalues of the Neumann Laplacian, when $n \geq 3$. The result fails for $n=2$, because the second eigenvalue is known to be minimal for the degenerate acute isosceles triangle (rather than for the equilateral) while the first eigenvalue is 0 for every triangle. We show the third eigenvalue is minimal for the equilateral triangle.