Except maybe the following fact : I think that the set of triangles such that is simple is open and dense
(and also full measure for a natural class of Lebesgue measure).
We have proved that for any mixed Dirichlet-Neumann boundary condition … except Neumann everywhere ! I have a sketch of proof
for the latter case but I never carried out the details (so there may be some bugs in the argument).

Last thing concerning analyticity of the eigenvalues and eigenfunctions, this holds only for one-parameter analytic families of triangles.
I don’t think the eigenvalues can be arranged to be analytic on the full parameter space (because there are crossings).

To get a triangle one can use
python eig.py tr a b -N -s number -m -c3 -e3
tr is domain specification, a,b is the third vertex, -N gives Neumann, -s number is number of triangles, -m shows mesh, -c3 gives contours instead of surface plots (3 contours are good for nodal line), -e3 to get 3 eigenvalues
There are many options. python eig.py -h lists all of them with minimalistic explanations

However, I think one does have analyticity as long as the extrema are unique (up to symmetry, in the isosceles case) and non-degenerate (i.e. their Hessian is definite), and the eigenvalue is simple. This is for instance the case for the non-equilateral acute isosceles and right-angled triangles, where we know that the eigenvalues are simple and the extrema only occur at the vertices of the longest side, and a Bessel expansion at a (necessarily acute) extremal vertex shows that any extremum is non-degenerate (it looks like a non-zero scalar multiple of the 0th Bessel function , plus lower order terms which are as ). Certainly in this setting, the work of Banuelos and Pang ( http://eudml.org/doc/130789;jsessionid=080D9E5423278BA5ACFC818847CA97FE ) applies, and small perturbations of the triangle give small perturbations of the eigenfunction in L^infty norm at least. This (together with uniform C^2 bounds for eigenfunctions in a compact family of acute triangles, which is sketched on the wiki, and is needed to handle the regions near the vertices) is already enough to give the hot spots conjecture for sufficiently small perturbations of a right-angled or non-equilateral acute isosceles triangle.

The Banuelos-Pang results require the eigenvalue to be simple, so the perturbation theory of the equilateral triangle (in which the second eigenvalue has multiplicity 2) is not directly covered. However, it seems very likely that for any sufficiently small perturbation of the equilateral triangle, a second eigenfunction of the perturbed triangle should be close in L^infty norm to _some_ second eigenfunction of the original triangle (but this approximating eigenfunction could vary quite discontinuously with respect to the perturbation). Assuming this, this shows the hot spots conjecture for perturbations of the equilateral triangle as well, because _every_ second eigenfunction of the equilateral triangle can be shown to have extrema only at the vertices, and to be uniformly bounded away from the extremum once one has a fixed distance away from the vertices (this comes from the strict concavity of the image of the complex second eigenfunction of the equilateral triangle, discussed on the wiki).

The perturbation argument also shows that in order for the hot spots conjecture to fail, there must exist a “threshold” counterexample of an acute triangle in which one of the vertex extrema is matched by a critical point either on the edge or interior of the triangle, though it is not clear to me how to use this information.

An analytic perturbation argument from known cases would certainly be great! I thought about a similar argument for the thin triangle case (http://michaelnielsen.org/polymath1/index.php?title=The_hot_spots_conjecture, under ‘thin not-quite-sectors). But I was thinking about perturbing from a sector to the triangle, and you’re thinking about perturbing from one triangle to another.

Let’s see if I follow your argument. Following the notation in (http://michaelnielsen.org/polymath1/index.php?title=The_hot_spots_conjecture, under ‘reformulation on a reference domain’), one can replace the reference triangle by any other. One then shows analyticity of the eigenvalues with respect to perturbations in the mapping B, and shows the domain of analyticity is large enough to cover all acute triangles. Is this correct?