Hot Spots Conjecture in Spectral Geometry

• The Hot Spots Conjecture asserts that the first nonconstant Neumann eigenfunction attains its extrema on the boundary of a domain.
• It bridges spectral geometry, probability, and variational approaches to analyze the behavior of eigenfunctions via heat flow dynamics.
• Recent studies provide sharp quantitative bounds on the hot spots ratio and delineate domain conditions and counterexamples influencing interior extrema.

The Hot Spots Conjecture is a conjecture in spectral geometry about the first non-constant Neumann eigenfunction of the Laplacian on a bounded domain. If $\Omega\subset \mathbb{R}^d$ is bounded and connected, and $u$ solves

$$-\Delta u=\mu_2 u \quad \text{in }\Omega, \qquad \partial_\nu u=0 \quad \text{on }\partial\Omega,$$

where $\mu_2$ is the smallest positive Neumann eigenvalue, then the conjecture predicts that the maxima and minima of $u$ occur on $\partial\Omega$, rather than in the interior. Since the Neumann ground state is constant, this is equivalently a statement about the second Neumann eigenfunction. In heat-flow language, it says that for an insulated body the asymptotically hottest and coldest points should lie on the boundary. The conjecture is classical, is attributed to Rauch, and remains a central organizing problem linking PDE, nodal geometry, probability, rearrangement inequalities, and geometric analysis (Steinerberger, 2021).

1. Spectral formulation and geometric meaning

For the Neumann Laplacian on a bounded domain, the spectrum is

$0=\mu_1<\mu_2\le \mu_3\le \cdots,$

with $\mu_1=0$ corresponding to constant eigenfunctions. The hot spots problem is therefore about eigenfunctions for $\mu_2$, the first nonzero Neumann eigenvalue. In its standard form, the conjecture asserts

$$\max_{x\in \Omega} u(x)=\max_{x\in \partial\Omega}u(x), \qquad \min_{x\in \Omega} u(x)=\min_{x\in \partial\Omega}u(x).$$

A stronger conclusion, available on some special domain classes, is that suitable directional derivatives of $u$0 have fixed sign in the interior; then the boundary-extremum statement follows from strict monotonicity. This stronger form has been established for higher-dimensional lip domains and certain symmetric domains, where after a suitable rotation every partial derivative satisfies a sign trichotomy: $u$1 throughout the domain (Kennedy et al., 2024).

The conjecture is naturally interpreted through the heat equation with Neumann boundary conditions. Long-time dynamics are controlled by the second Neumann mode, so the spatial extrema of $u$2 determine the eventual hot and cold spots. This spectral interpretation extends beyond Euclidean domains. It motivates mixed-boundary analogues, constant-curvature variants, manifold versions, and analogues on fractals and quantum graphs, although in those settings the precise meaning of “boundary” and of “hot spots” may differ from the Euclidean case (Hatcher, 2024).

2. Failure of the conjecture and sharp quantitative bounds

The conjecture is false in general. Counterexamples were constructed by Burdzy–Werner and Burdzy for domains with holes, showing that interior hot spots can occur on multiply connected planar domains (Steinerberger, 2021). Numerical work later exhibited easy-to-construct planar domains with one or more holes for which the first nonzero Neumann eigenfunction has interior extrema, and reported examples in which the ratio between an interior extremum and the corresponding boundary extremum exceeds $u$3 (Kleefeld, 2021).

A more recent development is that convexity does not rescue the conjecture in high dimension. There exists a sequence $u$4 of smooth, centrally symmetric convex domains in $u$5 such that

$u$6

and similarly for the minimum. Thus, for all sufficiently large $u$7, even smooth convex domains can have interior hot spots. The construction proceeds through a log-concave analogue of the conjecture and a high-dimensional “barrel domain” approximation of weighted problems by uniform measures on convex sets (Pont, 2024).

At the same time, failure is quantitatively limited. An earlier universal estimate proved that for every bounded, connected smooth domain $u$8, if $u$9 is the first nontrivial Neumann eigenfunction then

$$-\Delta u=\mu_2 u \quad \text{in }\Omega, \qquad \partial_\nu u=0 \quad \text{on }\partial\Omega,$$0

so the conjecture cannot fail by an arbitrary factor (Steinerberger, 2021). This has now been sharpened to an exact theorem. For a bounded, connected Lipschitz domain $$-\Delta u=\mu_2 u \quad \text{in }\Omega, \qquad \partial_\nu u=0 \quad \text{on }\partial\Omega,$$1, letting $$-\Delta u=\mu_2 u \quad \text{in }\Omega, \qquad \partial_\nu u=0 \quad \text{on }\partial\Omega,$$2 denote the first non-constant Neumann eigenfunction, the hot spots ratio

$$-\Delta u=\mu_2 u \quad \text{in }\Omega, \qquad \partial_\nu u=0 \quad \text{on }\partial\Omega,$$3

has extremal value

$$-\Delta u=\mu_2 u \quad \text{in }\Omega, \qquad \partial_\nu u=0 \quad \text{on }\partial\Omega,$$4

where $$-\Delta u=\mu_2 u \quad \text{in }\Omega, \qquad \partial_\nu u=0 \quad \text{on }\partial\Omega,$$5 is the radial solution of

$$-\Delta u=\mu_2 u \quad \text{in }\Omega, \qquad \partial_\nu u=0 \quad \text{on }\partial\Omega,$$6

The theorem gives

$$-\Delta u=\mu_2 u \quad \text{in }\Omega, \qquad \partial_\nu u=0 \quad \text{on }\partial\Omega,$$7

It also shows that no extremizer exists for $$-\Delta u=\mu_2 u \quad \text{in }\Omega, \qquad \partial_\nu u=0 \quad \text{on }\partial\Omega,$$8, and that near-maximizers must converge quantitatively to a ball in Fraenkel asymmetry (Pont et al., 22 Aug 2025).

The same work identifies a measure-theoretic form of asymptotic truth. If

$$-\Delta u=\mu_2 u \quad \text{in }\Omega, \qquad \partial_\nu u=0 \quad \text{on }\partial\Omega,$$9

then for every fixed $\mu_2$0,

$\mu_2$1

with exponential decay. Thus the maximal pointwise failure persists, since $\mu_2$2, but the set on which the eigenfunction significantly exceeds its boundary maximum occupies a vanishing fraction of the domain in high dimension (Pont et al., 22 Aug 2025).

3. Extremal mechanisms and comparison principles

The sharp upper bound for $\mu_2$3 is based on a comparison between the normalized Neumann eigenfunction and the positive solution of a Dirichlet boundary-value problem. With $\mu_2$4, one considers

$\mu_2$5

and proves the chain

$\mu_2$6

Here $\mu_2$7 denotes domination in the symmetric-decreasing rearrangement sense. The three inputs are a comparison principle, a Talenti-type rearrangement inequality, and monotonicity in $\mu_2$8 on the ball. Stability is then imported from the sharp Szegő–Weinberger inequality, yielding the conclusion that near-equality forces the domain to be close to a ball (Pont et al., 22 Aug 2025).

A notable feature of the sharp theorem is that the ball is the geometric limit of extremizing domains but is not itself a maximizer. Indeed, the ball has hot spots ratio $\mu_2$9, because its first nontrivial Neumann eigenfunction is not radial. The extremizing mechanism instead arises from domains that are “almost balls” but possess a weakly disconnected radial structure: an inner ball and outer shell connected by many thin channels. The first Neumann eigenfunction then becomes effectively radial and can develop a much larger interior maximum than its maximal boundary value (Pont et al., 22 Aug 2025).

Sharpness is realized through Neumann sieve domains. In the limit, these converge to an effective operator with jump penalty across a sphere,

$u$0

where $u$1 is shell thickness, $u$2 is connectivity strength, and $u$3 is the effective coupling. For suitable choices, the first eigenfunction of the effective problem converges to $u$4, producing the exact lower bound $u$5 (Pont et al., 22 Aug 2025).

4. Domain classes for which the conjecture is known to hold

Despite the general counterexamples, the conjecture is proved for many geometrically rigid classes. A major family is that of lip domains. In the plane, a new variational principle on vector fields shows that the first positive Neumann eigenvalue can be characterized by minimizing a quadratic form whose minimizers are gradients of eigenfunctions rather than the eigenfunctions themselves. On planar lip domains, this yields strict monotonicity along the two orthogonal directions $u$6 and $u$7, and therefore excludes interior extrema (Rohleder, 2021). The same framework was later presented as a broader variational approach connecting gradients of Neumann and Dirichlet eigenfunctions through an operator with curvature boundary term, and recovering known planar hot-spots results in a unified way (Rohleder, 2024).

In higher dimensions, the conjecture has been proved in a strong form for higher-dimensional lip domains and for certain symmetric domains generalizing the planar theorem of Jerison and Nadirashvili. The method uses a vector-valued Laplace operator whose spectrum contains the Neumann spectrum. This gives a deterministic, variational proof that avoids stochastic analysis and deformation arguments, and implies the boundary-extremum statement together with the sign trichotomy for each coordinate derivative (Kennedy et al., 2024).

Polygonal domains provide another major class. The conjecture holds for acute triangles with one angle not larger than $u$8, and for those triangles the second Neumann eigenvalue is simple when the triangle is non-equilateral (Siudeja, 2013). It also holds for L-shaped domains: second Neumann eigenfunctions on an L-shaped domain have no non-vertex critical points, their extrema occur only at the diametric vertices, and in the canonical embedding one can choose the eigenfunction so that

$u$9

This extends to five L-tiled non-convex polygonal families, including doubly connected domains (Hatcher, 2024).

The conjecture also holds for thin tubular neighborhoods of curves on surfaces. If $\partial\Omega$0 is a sufficiently thin strip around a smooth embedded curve on a smooth oriented $\partial\Omega$1-dimensional Riemannian manifold, then the second Neumann eigenfunction has no interior extrema, and more generally low-lying extrema can be located by comparison with one-dimensional Neumann eigenfunctions on the base curve (Krejcirik et al., 2017).

5. Curved spaces, mixed problems, and analogues

Several works extend the hot-spots principle beyond Euclidean Neumann domains. On constant-curvature surfaces, the conjecture holds for all non-acute geodesic triangles of constant negative curvature: a second Neumann eigenfunction has no critical points in the interior or on edges away from the vertices, and there exists a Killing field $\partial\Omega$2 such that $\partial\Omega$3 in the triangle. Under an additional mixed-boundary inequality, the same conclusion is conditional in the spherical case (Hatcher, 18 Aug 2025). A different hyperbolic result proves that on finite-area geodesically convex planar domains in $\partial\Omega$4, if the least positive Neumann eigenvalue lies in $\partial\Omega$5, then every corresponding eigenfunction has no interior critical points; in particular, bounded convex hyperbolic planar domains of sufficiently large area have second Neumann eigenfunctions without interior critical points (Hatcher, 20 May 2026).

Mixed Dirichlet–Neumann analogues display a related but distinct theory. For Euclidean triangles with one or two Dirichlet edges, first mixed eigenfunctions have at most one critical point, and that point lies on the Neumann part of the boundary; for broad classes of planar graph domains,

$\partial\Omega$6

so there are no interior critical points (Hatcher, 2024). On L-shaped domains and rectangles, several mixed hot-spots theorems have also been obtained and serve as key inputs for non-convex Neumann results (Hatcher, 2024).

The conjecture has further been extended to Riemannian settings with special local structure. On analytic Riemannian manifolds with isothermal coordinates, including hyperbolic spaces $\partial\Omega$7 and spheres $\partial\Omega$8, generalized lip domains and certain $\partial\Omega$9-symmetric domains satisfy the hot-spots conclusion. The method is Hodge-theoretic: one studies $0=\mu_1<\mu_2\le \mu_3\le \cdots,$0 as an eigenform for a mixed Hodge Laplacian, rather than differentiating the scalar equation directly (Hua et al., 2024). On compact warped products $0=\mu_1<\mu_2\le \mu_3\le \cdots,$1, second Neumann eigenfunctions have boundary extrema under radiality or suitable monotonicity and simplicity hypotheses on the warping function, while on infinite cones the analogous “hot spots” problem becomes a question about long-time heat-kernel asymptotics with four possible regimes controlled by the spectral gap of the fiber (Hatcher, 25 Aug 2025).

Analogue theories also exist in nonclassical settings. The conjecture is proved on the Vicsek set: every eigenfunction of the second smallest Neumann eigenvalue attains its maximum and minimum on the boundary $0=\mu_1<\mu_2\le \mu_3\le \cdots,$2 (Ionescu et al., 2018). On quantum graphs, the direct Euclidean boundary analogy fails, but the extremal points of $0=\mu_1<\mu_2\le \mu_3\le \cdots,$3-eigenfunctions are constrained: they can occur only at degree-one vertices or inside the doubly connected part of the graph, and their precise location depends strongly on edge lengths (Kennedy et al., 2020).

6. Methods, related programs, and current perspective

Three methodological strands recur throughout the literature. The first is probabilistic. Universal upper bounds were obtained from the heat equation, reflected Brownian motion, Dirichlet heat-kernel estimates, and eigenvalue comparison inequalities such as Faber–Krahn and Szegő–Weinberger (Steinerberger, 2021). The second is variational. Gradient-based formulations, vector-valued operators, and Hodge-theoretic decompositions convert the problem from one about scalar extrema to one about sign structure and monotonicity of derivatives or differential forms (Rohleder, 2021, Kennedy et al., 2024, Hua et al., 2024). The third is nodal-geometric. In polygonal and constant-curvature settings, corner expansions, reflection principles, and nodal-domain arguments are used to rule out critical points or to force them to the boundary [(Siudeja, 2013); (Hatcher, 2024); (Hatcher, 18 Aug 2025)].

A more programmatic direction views the conjecture as a level-set regularity problem. Jerison proposed a deformation-based approach in which one seeks Lipschitz control of level sets, especially the nodal set, and related this to a broader Two Hyperplane Conjecture for isoperimetric surfaces in centrally symmetric convex bodies. This perspective emphasizes large-scale connectivity and flatness of level sets rather than direct PDE maximum-principle arguments, but it remains a program rather than a solution (Jerison, 2018).

The conjecture also appears in applications as an assumption to be removed rather than proved. In low-frequency inference for reflected diffusions, earlier arguments had used a hot-spots-type nonvanishing-gradient property for a first Neumann eigenfunction. This dependence was later bypassed by replacing a single-eigenfunction argument with a finite family of eigenfunctions, showing that the conjecture is not needed for the relevant minimax and stability theorems (Alberti et al., 2024).

The present picture is therefore sharply bifurcated. The conjecture is false in full generality, and even false for convex smooth domains in sufficiently large dimension (Pont, 2024). Yet failure is quantitatively controlled, with an exact worst-case ratio in every dimension and a high-dimensional limit $0=\mu_1<\mu_2\le \mu_3\le \cdots,$4 (Pont et al., 22 Aug 2025). At the same time, a wide range of geometric settings still satisfy strong hot-spots theorems, often with monotonicity statements stronger than boundary attainment itself (Kennedy et al., 2024). A plausible overall interpretation is that the hot-spots phenomenon is not universal, but it is highly structured: when it fails, the failure is constrained by geometry, spectrum, and rearrangement; when it holds, it often does so through rigidity mechanisms that are visible in derivatives, nodal sets, or effective one-dimensional reductions.