Reilly Inequality & Its Extensions

• Reilly Inequality is an extrinsic spectral estimate that bounds the first nonzero Laplace–Beltrami eigenvalue via an averaged L2 norm of the mean curvature on a submanifold.
• It arises from Reilly’s integral formula, converting ambient curvature integrals into boundary terms and underpinning a range of geometric inequalities.
• Generalizations extend to weighted, nonlinear, and anisotropic cases, providing rigidity characterizations and eigenvalue bounds across diverse geometric settings.

Reilly inequality commonly denotes the extrinsic spectral estimate that bounds the first nonzero Laplace–Beltrami eigenvalue of a compact submanifold by an average of its squared mean curvature, and, more broadly, the family of Reilly-type inequalities derived from Reilly’s integral formula by changing the operator, the ambient geometry, or the underlying class of spaces. In one standard Euclidean normalization, for an isometric immersion $\phi:\Sigma^n\to \mathbb R^m$,

$$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$

(Kokarev, 2017). In another convention, for a compact $m$-dimensional submanifold $M^m\subset \mathbb R^N$,

$$\lambda_1^M \leqslant \frac{1}{m\,v_M}\int_M \|{\bf H}\|^2\,dv$$

(Grosjean et al., 11 Jul 2025). The subject now includes space-form, weighted, nonlinear, tensorial, differential-form, and singular generalizations.

1. Classical formulation and equality structure

In the classical closed-submanifold setting, Reilly’s inequality is an upper bound for the first nonzero Laplace eigenvalue in terms of extrinsic curvature. For an $n$-dimensional closed orientable submanifold $M$ in an $N$-dimensional space form $\mathbb M_c^N$, one standard statement is

$$\lambda_1 \le \frac{n}{\operatorname{vol}(M)}\int_M \big(|H|^2+c\big)\, dV_M$$

(Chen et al., 2018). In Euclidean space this reduces to the familiar mean-curvature bound. The literature records the estimate in several normalization conventions, but the geometric content is the same: the first nontrivial spectral frequency is controlled by an averaged $$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$0-curvature quantity (Kokarev, 2017, Grosjean et al., 11 Jul 2025).

The equality case is rigid. In one formulation, equality holds if and only if the submanifold is minimally immersed in a geodesic sphere of the ambient space form, with explicit radius determined by the eigenvalue (Chen et al., 2018). In the Euclidean hypersurface case, equality forces a round sphere; in higher codimension, equality forces, after scaling and Euclidean motion, an isometric minimal immersion into the unit sphere by first eigenfunctions (Kokarev, 2017). The varifold literature preserves the same pattern: equality means support on a sphere and, in codimension one, a constant-multiplicity sphere (Grosjean et al., 11 Jul 2025).

2. Reilly’s integral formula and the proof paradigm

The analytic source of the inequality is Reilly’s formula. For a compact manifold $$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$1 with smooth boundary $$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$2, with $$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$3 and $$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$4, one classical form is

$$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$5

(Qiu et al., 2014). This identity converts Bochner-type bulk terms into boundary geometry, and it underlies both eigenvalue bounds and geometric inequalities.

A major modern development is the weighted and curvature-shifted generalization. Qiu–Xia proved

$$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$6

which reduces to the classical formula when $$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$7 and $$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$8 (Qiu et al., 2014). With $$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$9, $m$0, or $m$1, $m$2, the bulk terms simplify in the hemisphere and hyperbolic space, yielding alternative proofs of Alexandrov and Heintze–Karcher type results (Qiu et al., 2014).

The same strategy appears in geometric inequalities. Xia solved a weighted Neumann problem and inserted it into the generalized Reilly identity to obtain the optimal Minkowski-type inequality

$m$3

in $m$4 and $m$5, under the boundary condition

$m$6

(Xia, 2015). In capillary problems, a Reilly-type formula on domains with piecewise smooth boundary, combined with a mixed boundary value problem tailored to the contact angle, yields Heintze–Karcher inequalities and Alexandrov-type rigidity for capillary hypersurfaces in a half-space or half-ball (Jia et al., 2022).

3. Nonlinear, anisotropic, and tensorial generalizations

A large part of the modern theory replaces the Laplacian by other elliptic operators while keeping the extrinsic structure of the bound. For the $m$7-Laplacian on closed orientable submanifolds in space forms, Chen–Wei derived upper bounds for $m$8 in terms of $m$9, with one regime for $M^m\subset \mathbb R^N$0 and another for $M^m\subset \mathbb R^N$1; equality occurs only when $M^m\subset \mathbb R^N$2, in which case the submanifold is minimally immersed in a geodesic sphere (Chen et al., 2018). In Cartan–Hadamard manifolds with $M^m\subset \mathbb R^N$3, analogous Reilly-type estimates were obtained for both $M^m\subset \mathbb R^N$4 and the first eigenvalue of $M^m\subset \mathbb R^N$5, improving earlier supremum-type estimates to integral ones (Chen et al., 2022).

For divergence-form operators, Chen–Wang considered

$M^m\subset \mathbb R^N$6

where $M^m\subset \mathbb R^N$7 is symmetric, positive definite, and divergence-free, and defined

$M^m\subset \mathbb R^N$8

Their sharp bound is

$M^m\subset \mathbb R^N$9

under the semipositivity assumption $$\lambda_1^M \leqslant \frac{1}{m\,v_M}\int_M \|{\bf H}\|^2\,dv$$0 (Chen et al., 2018). Equality is characterized by constancy of $$\lambda_1^M \leqslant \frac{1}{m\,v_M}\int_M \|{\bf H}\|^2\,dv$$1 and $$\lambda_1^M \leqslant \frac{1}{m\,v_M}\int_M \|{\bf H}\|^2\,dv$$2-minimality in a geodesic sphere; with $$\lambda_1^M \leqslant \frac{1}{m\,v_M}\int_M \|{\bf H}\|^2\,dv$$3, this recovers the classical Reilly inequality (Chen et al., 2018). In the negatively curved Cartan–Hadamard setting, the corresponding estimate becomes

$$\lambda_1^M \leqslant \frac{1}{m\,v_M}\int_M \|{\bf H}\|^2\,dv$$4

(Chen et al., 2022).

Anisotropic versions also exist. Given a positive $$\lambda_1^M \leqslant \frac{1}{m\,v_M}\int_M \|{\bf H}\|^2\,dv$$5 on $$\lambda_1^M \leqslant \frac{1}{m\,v_M}\int_M \|{\bf H}\|^2\,dv$$6 satisfying a convexity condition, the $$\lambda_1^M \leqslant \frac{1}{m\,v_M}\int_M \|{\bf H}\|^2\,dv$$7-th anisotropic mean curvature $$\lambda_1^M \leqslant \frac{1}{m\,v_M}\int_M \|{\bf H}\|^2\,dv$$8 and the linearized operator $$\lambda_1^M \leqslant \frac{1}{m\,v_M}\int_M \|{\bf H}\|^2\,dv$$9 are defined for hypersurfaces in $n$0, and Reilly type inequalities for the first eigenvalue of $n$1 are proved (He, 2011).

4. Weighted and ambient-curvature variants

A second major direction replaces the unweighted Laplacian by a drift or weighted Laplacian and replaces ordinary mean curvature by a density-corrected object. For a compact submanifold $n$2 with density $n$3, the weighted Laplacian is

$n$4

and the weighted mean curvature vector is

$n$5

The resulting Reilly-type estimate is

$n$6

and equality implies that the density is Gaussian along $n$7 and that $n$8 is a self-shrinker for mean curvature flow (Domingo-Juan et al., 2015). In Euclidean space and on spheres, the equality analysis identifies Gaussian densities and shrinkers as the weighted analogues of the classical spherical equality objects (Domingo-Juan et al., 2015).

For weighted $n$9-Laplacians on manifolds with boundary, one has a Reilly-type integral formula associated with

$M$0

together with weighted boundary quantities $M$1 and $M$2. This framework yields weighted Heintze–Karcher and Minkowski inequalities, almost Schur lemmas, and eigenvalue relations for Wentzell and Steklov problems (Huang et al., 2022). In particular, under $M$3 and $M$4,

$M$5

while under $M$6 and $M$7,

$M$8

(Huang et al., 2022).

The same weighted philosophy appears in ambient-curvature comparison results. In manifolds with boundary and $M$9, Qiu–Xia obtained

$N$0

with $N$1, and equality holds if and only if $N$2 is a geodesic ball in a space form of sectional curvature $N$3 (Qiu et al., 2014). This sits alongside the space-form Minkowski inequality of Xia (Xia, 2015).

5. Differential forms, weighted forms, and singular spaces

Reilly theory for differential forms begins with the Raulot–Savo formula. If $N$4 is compact with smooth boundary $N$5, and $N$6 while the $N$7-curvatures satisfy $N$8, then the first eigenvalue of the Hodge Laplacian on exact $N$9-forms satisfies

$\mathbb M_c^N$0

for $\mathbb M_c^N$1 (Raulot et al., 2010). In the Euclidean ball this is sharp, and under nonnegative Ricci curvature equality rigidly characterizes the Euclidean ball (Raulot et al., 2010). The scalar Reilly identity is recovered by taking $\mathbb M_c^N$2 in the differential-form formula (Raulot et al., 2010).

Weighted differential-form analogues have recently been developed on smooth metric measure spaces. A weighted Reilly type integral formula for differential forms on a compact smooth metric measure space with boundary leads to lower bounds for the weighted Hodge Laplacian on the boundary, vanishing and rigidity statements for absolute cohomology, and lower bounds for Steklov eigenvalues of forms (Liyi et al., 8 Dec 2025). A parallel development on weighted manifolds derives a weighted Reilly formula for differential forms, a Poincaré-type inequality, weighted boundary value problems, and new eigenvalue estimates (Chami et al., 5 Dec 2025).

The Reilly method also extends to generalized geometric measure settings. For a rectifiable $\mathbb M_c^N$3 varifold $\mathbb M_c^N$4 of mass $\mathbb M_c^N$5, the first Rayleigh-type spectral quantity is defined by

$\mathbb M_c^N$6

and satisfies

$\mathbb M_c^N$7

Equality forces the support to lie on a sphere; in codimension one, the varifold is exactly a constant-multiplicity round sphere (Grosjean et al., 11 Jul 2025). The same paper proves a polygonal analogue,

$\mathbb M_c^N$8

with a richer equality class consisting of regular polygons, rhombi, trapezoids, and fake-regular polygons (Grosjean et al., 11 Jul 2025).

6. Higher eigenvalues, flow-related variants, and terminology

The classical inequality concerns the first nonzero eigenvalue, but higher-eigenvalue analogues are now established. Kokarev proved that if $\mathbb M_c^N$9 is an isometric immersion of a closed manifold, then for every $$\lambda_1 \le \frac{n}{\operatorname{vol}(M)}\int_M \big(|H|^2+c\big)\, dV_M$$0,

$$\lambda_1 \le \frac{n}{\operatorname{vol}(M)}\int_M \big(|H|^2+c\big)\, dV_M$$1

and more generally

$$\lambda_1 \le \frac{n}{\operatorname{vol}(M)}\int_M \big(|H|^2+c\big)\, dV_M$$2

for immersions into ambient manifolds conformally immersible into a sphere (Kokarev, 2017). Earlier universal inequalities also led to multi-eigenvalue Reilly-type bounds for sums of Laplace eigenvalues on Euclidean submanifolds (Ilias et al., 2010).

The term also appears in nonlinear one-dimensional geometric analysis. For closed immersed curves $$\lambda_1 \le \frac{n}{\operatorname{vol}(M)}\int_M \big(|H|^2+c\big)\, dV_M$$3, the ratio

$$\lambda_1 \le \frac{n}{\operatorname{vol}(M)}\int_M \big(|H|^2+c\big)\, dV_M$$4

admits a sharp positive lower bound on every sublevel $$\lambda_1 \le \frac{n}{\operatorname{vol}(M)}\int_M \big(|H|^2+c\big)\, dV_M$$5, while the corresponding infimum on $$\lambda_1 \le \frac{n}{\operatorname{vol}(M)}\int_M \big(|H|^2+c\big)\, dV_M$$6 is $$\lambda_1 \le \frac{n}{\operatorname{vol}(M)}\int_M \big(|H|^2+c\big)\, dV_M$$7 (Müller et al., 2019). This is explicitly called a constrained sharp Reilly-type inequality and is used to control length along the elastic flow and prove convergence below the critical energy $$\lambda_1 \le \frac{n}{\operatorname{vol}(M)}\int_M \big(|H|^2+c\big)\, dV_M$$8 (Müller et al., 2019).

A common misconception is terminological. Reilly inequalities belong to spectral geometry and geometric integral formulas, whereas Rellich inequalities are higher-order Hardy-type inequalities involving $$\lambda_1 \le \frac{n}{\operatorname{vol}(M)}\int_M \big(|H|^2+c\big)\, dV_M$$9, $$\lambda_1(\Sigma^n,g)\le \frac{n}{\operatorname{Vol}_g(\Sigma^n)}\int_{\Sigma^n}|H_\phi|^2\,d\operatorname{Vol}_g$$00, and singular weights. Papers such as “Rellich Inequality via Radial Dissipativity” and “Rellich inequalities via Riccati pairs on model space forms” concern Rellich, not Reilly, theory (Huang, 2023, Kajántó, 2023). The distinction is structural: Reilly theory is built around Bochner–Reilly identities, boundary geometry, and extrinsic spectral bounds; Rellich theory is built around second-order coercive inequalities with singular weights.

Across these variants, the unifying pattern is stable. Reilly-type statements compare a spectral quantity or geometric quotient with an averaged curvature term; equality is typically rigid and sphere-like; and the integral formula remains the organizing device, even when the setting has changed to weighted manifolds, nonlinear operators, differential forms, varifolds, or capillary boundary geometries.