The previous research thread for the Polymath7 project “the Hot Spots Conjecture” is now quite full, so I am now rolling it over to a fresh thread both to summarise the progress thus far, and to make it a bit easier to catch up on the latest developments.

The objective of this project is to prove that for an acute angle triangle ABC, that

- The second eigenvalue of the Neumann Laplacian is simple (unless ABC is equilateral); and
- For any second eigenfunction of the Neumann Laplacian, the extremal values of this eigenfunction are only attained on the boundary of the triangle. (Indeed, numerics suggest that the extrema are only attained at the corners of a side of maximum length.)

To describe the progress so far, it is convenient to draw the following “map” of the parameter space. Observe that the conjecture is invariant with respect to dilation and rigid motion of the triangle, so the only relevant parameters are the three angles of the triangle. We can thus represent any such triangle as a point in the region . The parameter space is then the following two-dimensional triangle:

Thus, for instance

- A,N,P represent the degenerate obtuse triangles (with two angles zero, and one angle of 180 degrees);
- B,F,O represent the degenerate acute isosceles triangles (with two angles 90 degrees, and one angle zero);
- C,E,G,I,L,M represent the various permutations of the 30-60-90 right-angled triangle;
- D,J,K represent the isosceles right-angled triangles (i.e. the 45-45-90 triangles);
- H represents the equilateral triangle (i.e. the 60-60-60 triangle);
- The acute triangles form the interior of the region BFO, with the edges of that region being the right-angled triangles, and the exterior being the obtuse triangles;
- The isosceles triangles form the three line segments NF, BP, AO. Sub-equilateral isosceles triangles (with apex angle smaller than 60 degrees) comprise the open line segments BH,FH,OH, while super-equilateral isosceles triangles (with apex angle larger than 60 degrees) comprise the complementary line segments AH, NH, PH.

Of course, one could quotient out by permutations and only work with one sixth of this diagram, such as ABH (or even BDH, if one restricted to the acute case), but I like seeing the symmetry as it makes for a nicer looking figure.

Here’s what we know so far with regards to the hot spots conjecture:

- For obtuse or right-angled triangles (the blue shaded region in the figure), the monotonicity results of Banuelos and Burdzy show that the second claim of the hot spots conjecture is true for at least one second eigenfunction.
- For any isosceles non-equilateral triangle, the eigenvalue bounds of Laugesen and Siudeja show that the second eigenvalue is simple (i.e. the first part of the hot spots conjecture), with the second eigenfunction being symmetric around the axis of symmetry for sub-equilateral triangles and anti-symmetric for super-equilateral triangles.
- As a consequence of the above two facts and a reflection argument found in the previous research thread, this gives the second part of the hot spots conjecture for sub-equilateral triangles (the green line segments in the figure). In this case, the extrema only occur at the vertices.
- For equilateral triangles (H in the figure), the eigenvalues and eigenfunctions can be computed exactly; the second eigenvalue has multiplicity two, and all eigenfunctions have extrema only at the vertices.
- For sufficiently thin acute triangles (the purple regions in the figure), the eigenfunctions are almost parallel to the sector eigenfunction given by the zeroth Bessel function; this in particular implies that they are simple (since otherwise there would be a second eigenfunction orthogonal to the sector eigenfunction). Also, a more complicated argument found in the previous research thread shows in this case that the extrema can only occur either at the pointiest vertex, or on the opposing side.

So, as the figure shows, there has been some progress on the problem, but there are still several regions of parameter space left to eliminate. It may be possible to use perturbation arguments to extend validity of the hot spots conjecture beyond the known regions by some quantitative extent, and then use numerical verification to finish off the remainder. (It appears that numerics work well for acute triangles once one has moved away from the degenerate cases B,F,O.)

The figure also suggests some possible places to focus attention on, such as:

- Super-equilateral acute triangles (the line segments DH, GH, KH). Here, we know the second eigenfunction is simple (and anti-symmetric).
- Nearly equilateral triangles (the region near H). The perturbation theory for the equilateral triangle could be non-trivial due to the repeated eigenvalue here.
- Nearly isosceles right-angled triangles (the regions near D,G,K). Again, the eigenfunction theory for isosceles right-angled triangles is very explicit, but this time the eigenvalue is simple and perturbation theory should be relatively straightforward.
- Nearly 30-60-90 triangles (the regions near C,E,G,I,L,M). Again, we have an explicit simple eigenfunction in the 30-60-90 case and an analysis should not be too difficult.

There are a number of stretching techniques (such as in the Laugesen-Siudeja paper) which are good for controlling how eigenvalues deform with respect to perturbations, and this may allow us to rigorously establish the first part of the hot spots conjecture, at least, for larger portions of the parameter space.

As for numerical verification of the second part of the conjecture, it appears that we have good finite element methods that seem to give accurate results in practice, but it remains to find a way to generate rigorous guarantees of accuracy and stability with respect to perturbations. It may be best to focus on the super-equilateral acute isosceles case first, as there is now only one degree of freedom in the parameter space (the apex angle, which can vary between 60 and 90 degrees) and also a known anti-symmetry in the eigenfunction, both of which should cut down on the numerical work required.

I may have missed some other points in the above summary; please feel free to add your own summaries or other discussion below.