Unweighted Yamabe-type Constants

• Unweighted Yamabe-type constants are conformal invariants derived from the scalar curvature energy and Sobolev quotients on Riemannian manifolds.
• They provide explicit model-space thresholds for surgery theory and compactness criteria in closed, noncompact, and boundary settings.
• Extensions to equivariant, higher-eigenvalue, spinorial, and CR settings offer nuanced insights into nodal solutions, symmetry restrictions, and geometric rigidity.

Unweighted Yamabe-type constants are conformal variational invariants defined from the standard scalar-curvature energy, the usual critical Sobolev exponents, and the ordinary Riemannian volume measure, with no auxiliary density or weighted curvature terms. In the closed case they include the classical Yamabe constant of a conformal class and the Yamabe invariant of a manifold; in noncompact, boundary, singular, equivariant, higher-eigenvalue, spinorial, and CR settings they appear as closely related infima of the same basic Sobolev-type quotient or of directly analogous unweighted functionals (Petean et al., 2012).

1. Classical definition and basic structure

For a closed $n$-manifold $M^n$ with conformal class $[g]$, the classical unweighted Yamabe constant is

$$Y(M,[g])=\inf_{h\in[g]}\frac{\int_M s_h\,d\operatorname{vol}(h)}{\operatorname{Vol}(M,h)^{\frac{n-2}{n}}},$$

where $s_h$ is the scalar curvature and $d\operatorname{vol}(h)$ is the Riemannian volume element. Writing $h=f^{p-2}g$ with

$$p=p_n=\frac{2n}{n-2},\qquad a_n=\frac{4(n-1)}{n-2},$$

this becomes the standard unweighted Yamabe functional

$$Y(M,[g])=\inf_{f\in C^\infty(M)} \frac{\displaystyle \int_M a_n|\nabla f|^2\,d\operatorname{vol}(g)+\int_M s_g f^2\,d\operatorname{vol}(g)} {\left(\displaystyle\int_M |f|^p\,d\operatorname{vol}(g)\right)^{2/p}}.$$

The corresponding manifold invariant is

$Y(M)=\sup_{[g]}Y(M,[g]).$

These are the prototype unweighted Yamabe-type constants: the measure is the ordinary Riemannian volume, and there is no density factor or modified scalar curvature (Petean et al., 2012).

Equivalent formulations occur throughout the literature. On a complete $M^n$0-manifold $M^n$1, one may write the conformal Laplacian

$M^n$2

and define

$M^n$3

with the test-function infimum

$M^n$4

On closed manifolds, the solution of the Yamabe problem implies $M^n$5, and for the round sphere one has

$M^n$6

This is the same unweighted conformal geometry written in operator form (Ammann et al., 2015).

A recurrent structural fact is Aubin’s inequality: for any closed $M^n$7-manifold $M^n$8,

$M^n$9

with equality on the round sphere. Several papers in the record use $[g]$0 as the normalization benchmark for lower bounds, comparison results, and gap phenomena (Petean et al., 2012).

2. Closed, noncompact, and boundary variants

For noncompact manifolds of positive scalar curvature, the same unweighted quotient is used with Sobolev test functions. If $[g]$1 is noncompact,

$[g]$2

This is the definition used for product spaces such as $[g]$3 and for noncompact model geometries appearing in surgery theory and adiabatic limits (Petean et al., 2012).

A more systematic noncompact treatment defines, for every metric $[g]$4 on a noncompact $[g]$5-manifold,

$[g]$6

where

$[g]$7

This quantity is conformally invariant. For noncompact $[g]$8, Kim’s Yamabe constant at infinity is

$[g]$9

independent of the compact exhaustion $$Y(M,[g])=\inf_{h\in[g]}\frac{\int_M s_h\,d\operatorname{vol}(h)}{\operatorname{Vol}(M,h)^{\frac{n-2}{n}}},$$0, and satisfies

$$Y(M,[g])=\inf_{h\in[g]}\frac{\int_M s_h\,d\operatorname{vol}(h)}{\operatorname{Vol}(M,h)^{\frac{n-2}{n}}},$$1

If $$Y(M,[g])=\inf_{h\in[g]}\frac{\int_M s_h\,d\operatorname{vol}(h)}{\operatorname{Vol}(M,h)^{\frac{n-2}{n}}},$$2, then $$Y(M,[g])=\inf_{h\in[g]}\frac{\int_M s_h\,d\operatorname{vol}(h)}{\operatorname{Vol}(M,h)^{\frac{n-2}{n}}},$$3 (Große et al., 2012).

On manifolds with boundary, the unweighted analogue is the relative Yamabe constant. If $$Y(M,[g])=\inf_{h\in[g]}\frac{\int_M s_h\,d\operatorname{vol}(h)}{\operatorname{Vol}(M,h)^{\frac{n-2}{n}}},$$4 is a conformal class on a compact manifold $$Y(M,[g])=\inf_{h\in[g]}\frac{\int_M s_h\,d\operatorname{vol}(h)}{\operatorname{Vol}(M,h)^{\frac{n-2}{n}}},$$5 with $$Y(M,[g])=\inf_{h\in[g]}\frac{\int_M s_h\,d\operatorname{vol}(h)}{\operatorname{Vol}(M,h)^{\frac{n-2}{n}}},$$6, and

$$Y(M,[g])=\inf_{h\in[g]}\frac{\int_M s_h\,d\operatorname{vol}(h)}{\operatorname{Vol}(M,h)^{\frac{n-2}{n}}},$$7

then

$$Y(M,[g])=\inf_{h\in[g]}\frac{\int_M s_h\,d\operatorname{vol}(h)}{\operatorname{Vol}(M,h)^{\frac{n-2}{n}}},$$8

Fixing $$Y(M,[g])=\inf_{h\in[g]}\frac{\int_M s_h\,d\operatorname{vol}(h)}{\operatorname{Vol}(M,h)^{\frac{n-2}{n}}},$$9, one has the Rayleigh-type characterization

$s_h$0

and minimizers satisfy the Yamabe-type PDE with Neumann boundary condition

$s_h$1

This is again completely unweighted; the only boundary restriction is minimality $s_h$2 (Hamanaka, 2020).

A dynamic version appears under Ricci flow with boundary. For a flow $s_h$3 with

$s_h$4

the relative Yamabe constant $s_h$5 is differentiated by first studying subcritical functionals $s_h$6. If the initial metric is a relative Yamabe metric and a $s_h$7-family of relative Yamabe metrics exists, then

$s_h$8

with equality if and only if $s_h$9 is Einstein (Botvinnik et al., 2019).

3. Model spaces, surgery thresholds, and explicit product estimates

A major role of unweighted Yamabe-type constants is to provide explicit model-space thresholds for surgery and bordism. One family is

$d\operatorname{vol}(h)$0

with metric $d\operatorname{vol}(h)$1. On these spaces one defines the scalar constants

$d\operatorname{vol}(h)$2

and from them the threshold quantities

$d\operatorname{vol}(h)$3

together with

$d\operatorname{vol}(h)$4

These are the scalar surgery thresholds in the theorem

$d\operatorname{vol}(h)$5

for a surgery of codimension $d\operatorname{vol}(h)$6. In the good range, one has $d\operatorname{vol}(h)$7, so the threshold is determined by the standard unweighted conformal Laplacian on the model spaces (Ammann et al., 2015).

The paper on square-integrability of solutions of the Yamabe equation sharpens this mechanism on

$d\operatorname{vol}(h)$8

If a smooth positive solution $d\operatorname{vol}(h)$9 of

$h=f^{p-2}g$0

satisfies

$h=f^{p-2}g$1

then $h=f^{p-2}g$2. This yields

$h=f^{p-2}g$3

for $h=f^{p-2}g$4, and also for $h=f^{p-2}g$5 in dimensions $h=f^{p-2}g$6. As a consequence, the surgery constants can be estimated by explicit unweighted model constants, producing bounds such as

$h=f^{p-2}g$7

for 2-connected compact $h=f^{p-2}g$8-manifolds, and

$h=f^{p-2}g$9

for 2-connected compact $$p=p_n=\frac{2n}{n-2},\qquad a_n=\frac{4(n-1)}{n-2},$$0-manifolds with vanishing $$p=p_n=\frac{2n}{n-2},\qquad a_n=\frac{4(n-1)}{n-2},$$1-index (Ammann et al., 2011).

Another prominent use of unweighted Yamabe-type constants is on products $$p=p_n=\frac{2n}{n-2},\qquad a_n=\frac{4(n-1)}{n-2},$$2. By comparing isoperimetric profiles with those of round spheres and then applying spherical symmetrization, one obtains explicit lower bounds. In dimension $$p=p_n=\frac{2n}{n-2},\qquad a_n=\frac{4(n-1)}{n-2},$$3,

$$p=p_n=\frac{2n}{n-2},\qquad a_n=\frac{4(n-1)}{n-2},$$4

numerically about $$p=p_n=\frac{2n}{n-2},\qquad a_n=\frac{4(n-1)}{n-2},$$5, and hence

$$p=p_n=\frac{2n}{n-2},\qquad a_n=\frac{4(n-1)}{n-2},$$6

for every closed surface $$p=p_n=\frac{2n}{n-2},\qquad a_n=\frac{4(n-1)}{n-2},$$7 (Petean et al., 2010).

In dimension $$p=p_n=\frac{2n}{n-2},\qquad a_n=\frac{4(n-1)}{n-2},$$8, the same isoperimetric-profile method yields

$$p=p_n=\frac{2n}{n-2},\qquad a_n=\frac{4(n-1)}{n-2},$$9

and also

$$Y(M,[g])=\inf_{f\in C^\infty(M)} \frac{\displaystyle \int_M a_n|\nabla f|^2\,d\operatorname{vol}(g)+\int_M s_g f^2\,d\operatorname{vol}(g)} {\left(\displaystyle\int_M |f|^p\,d\operatorname{vol}(g)\right)^{2/p}}.$$0

The key comparison theorem states that if

$$Y(M,[g])=\inf_{f\in C^\infty(M)} \frac{\displaystyle \int_M a_n|\nabla f|^2\,d\operatorname{vol}(g)+\int_M s_g f^2\,d\operatorname{vol}(g)} {\left(\displaystyle\int_M |f|^p\,d\operatorname{vol}(g)\right)^{2/p}}.$$1

and $$Y(M,[g])=\inf_{f\in C^\infty(M)} \frac{\displaystyle \int_M a_n|\nabla f|^2\,d\operatorname{vol}(g)+\int_M s_g f^2\,d\operatorname{vol}(g)} {\left(\displaystyle\int_M |f|^p\,d\operatorname{vol}(g)\right)^{2/p}}.$$2, then

$$Y(M,[g])=\inf_{f\in C^\infty(M)} \frac{\displaystyle \int_M a_n|\nabla f|^2\,d\operatorname{vol}(g)+\int_M s_g f^2\,d\operatorname{vol}(g)} {\left(\displaystyle\int_M |f|^p\,d\operatorname{vol}(g)\right)^{2/p}}.$$3

These estimates were positioned as complements to the explicit gap theorems of Ammann–Dahl–Humbert (Petean et al., 2012).

A distinct product asymptotic appears for higher Yamabe constants. If $$Y(M,[g])=\inf_{f\in C^\infty(M)} \frac{\displaystyle \int_M a_n|\nabla f|^2\,d\operatorname{vol}(g)+\int_M s_g f^2\,d\operatorname{vol}(g)} {\left(\displaystyle\int_M |f|^p\,d\operatorname{vol}(g)\right)^{2/p}}.$$4 has positive scalar curvature and $$Y(M,[g])=\inf_{f\in C^\infty(M)} \frac{\displaystyle \int_M a_n|\nabla f|^2\,d\operatorname{vol}(g)+\int_M s_g f^2\,d\operatorname{vol}(g)} {\left(\displaystyle\int_M |f|^p\,d\operatorname{vol}(g)\right)^{2/p}}.$$5 is closed, then

$$Y(M,[g])=\inf_{f\in C^\infty(M)} \frac{\displaystyle \int_M a_n|\nabla f|^2\,d\operatorname{vol}(g)+\int_M s_g f^2\,d\operatorname{vol}(g)} {\left(\displaystyle\int_M |f|^p\,d\operatorname{vol}(g)\right)^{2/p}}.$$6

and, when $$Y(M,[g])=\inf_{f\in C^\infty(M)} \frac{\displaystyle \int_M a_n|\nabla f|^2\,d\operatorname{vol}(g)+\int_M s_g f^2\,d\operatorname{vol}(g)} {\left(\displaystyle\int_M |f|^p\,d\operatorname{vol}(g)\right)^{2/p}}.$$7 is constant,

$$Y(M,[g])=\inf_{f\in C^\infty(M)} \frac{\displaystyle \int_M a_n|\nabla f|^2\,d\operatorname{vol}(g)+\int_M s_g f^2\,d\operatorname{vol}(g)} {\left(\displaystyle\int_M |f|^p\,d\operatorname{vol}(g)\right)^{2/p}}.$$8

For $$Y(M,[g])=\inf_{f\in C^\infty(M)} \frac{\displaystyle \int_M a_n|\nabla f|^2\,d\operatorname{vol}(g)+\int_M s_g f^2\,d\operatorname{vol}(g)} {\left(\displaystyle\int_M |f|^p\,d\operatorname{vol}(g)\right)^{2/p}}.$$9, this produces nodal solutions of the Yamabe equation on $Y(M)=\sup_{[g]}Y(M,[g]).$0 for large $Y(M)=\sup_{[g]}Y(M,[g]).$1 (Henry, 2015).

A recent gap theorem for products with small tori gives a complementary phenomenon. If $Y(M)=\sup_{[g]}Y(M,[g]).$2 is a Yamabe metric with $Y(M)=\sup_{[g]}Y(M,[g]).$3, then there exists $Y(M)=\sup_{[g]}Y(M,[g]).$4 such that whenever the flat torus metric $Y(M)=\sup_{[g]}Y(M,[g]).$5 satisfies $Y(M)=\sup_{[g]}Y(M,[g]).$6, the product metric $Y(M)=\sup_{[g]}Y(M,[g]).$7 on $Y(M)=\sup_{[g]}Y(M,[g]).$8 is also a Yamabe metric. The method extends to Type I and Type II Yamabe constants on manifolds with boundary (Wang et al., 24 May 2026).

4. Local, singular, and noncompact compactness thresholds

A persistent theme is that unweighted Yamabe-type constants behave as local compactness thresholds. On stratified spaces, for any open set $Y(M)=\sup_{[g]}Y(M,[g]).$9, one defines the local Sobolev and Yamabe constants

$M^n$00

$M^n$01

The local invariants are then

$M^n$02

Under mild scalar-curvature assumptions, $M^n$03, and if

$M^n$04

then the Yamabe problem admits a minimizer. On stratified spaces the local model constants are explicitly identified with Yamabe constants of tangent-model spaces such as $M^n$05 or conformally $M^n$06 (Akutagawa et al., 2012).

A 2024 local method for compact and noncompact Yamabe problems also hinges on comparison with the sphere. For a small Riemannian domain $M^n$07, $M^n$08, with $M^n$09 on $M^n$10 and nonvanishing Weyl tensor, Aubin-type test functions $M^n$11 yield a local quotient satisfying

$M^n$12

where $M^n$13 is the best Euclidean Sobolev constant and

$M^n$14

This gives positive solutions of local Dirichlet Yamabe-type equations and, after a super-local correction argument, test functions $M^n$15 with

$M^n$16

For compact manifolds this local inequality implies $M^n$17, and for certain complete noncompact manifolds pointwise conformal to subdomains of compact manifolds it implies

$M^n$18

thereby yielding solutions of the Yamabe equation in the positive case (Xu, 2024).

On general noncompact manifolds, the continuity theory of the unweighted Yamabe constant is itself governed by compactness at infinity. The map

$M^n$19

is continuous in the fine $M^n$20-topology, while the Yamabe constant at infinity

$M^n$21

is locally constant in that topology. By contrast, on noncompact $M^n$22 the map $M^n$23 is not continuous in any compact-open $M^n$24-topology at metrics with finite value (Große et al., 2012).

5. Symmetry restrictions, higher eigenvalues, and related analogues

Unweighted Yamabe-type constants admit several symmetry-restricted and higher-level variants. For a compact Lie group $M^n$25 acting on a closed manifold $M^n$26, the $M^n$27-equivariant Yamabe constant is

$M^n$28

where

$M^n$29

and the equivariant Yamabe invariant is

$M^n$30

For the Hopf $M^n$31-action on $M^n$32,

$M^n$33

More generally, for the action $M^n$34 on $M^n$35, if $M^n$36,

$M^n$37

For closed oriented 3-manifolds with free $M^n$38-action and quotient orbifold $M^n$39, one has the topological upper bound

$M^n$40

when $M^n$41 and $M^n$42, while $M^n$43 if $M^n$44 (Ammann et al., 2015).

Higher-eigenvalue constants are encoded by the $M^n$45-th Yamabe constants

$M^n$46

The case $M^n$47 is especially important because it is tied to nodal solutions of the Yamabe equation; if $M^n$48 is attained by a generalized metric on a connected manifold, the minimizer is sign-changing (Henry, 2015).

Spinorial counterparts are also unweighted. On a spin manifold $M^n$49, with $M^n$50, one defines

$M^n$51

for compactly supported spinors with $M^n$52, leading to

$M^n$53

After renormalization,

$M^n$54

On the model spaces $M^n$55, the noncompact conformal Hijazi inequality gives

$M^n$56

and hence

$M^n$57

These thresholds govern spinorial surgery monotonicity and spin bordism invariance below $M^n$58 (Ammann et al., 2015).

A different analogue arises in CR geometry. For a compact strongly pseudoconvex CR manifold $M^n$59, the CR Yamabe constant is

$M^n$60

and if $M^n$61, then

$M^n$62

This is explicitly described as the unweighted CR analogue of the classical Yamabe constant. When $M^n$63,

$M^n$64

for $M^n$65, and the paper constructs compact simply connected manifolds admitting two strongly pseudoconvex CR structures with different signs of the CR Yamabe constant (Sung et al., 2022).

6. Conceptual role and relation to weighted theories

The common feature of these invariants is that they are built from the standard conformal Laplacian, the standard Dirac operator, the standard boundary conformal Laplacian, or analogous unweighted subelliptic operators, always paired with the ordinary geometric measure. Several papers make this contrast explicit. In the smooth metric measure-space framework, the weighted Yamabe constant $M^n$66 reduces exactly to the classical Yamabe constant when $M^n$67, because then $M^n$68, $M^n$69, the weighted conformal Laplacian becomes

$M^n$70

and the quotient collapses to the usual Yamabe functional. The weighted theory is presented as an interpolation between the classical Yamabe problem at $M^n$71 and Perelman’s $M^n$72-entropy as $M^n$73 [(Posso, 2017); (Case, 2013)].

This suggests a useful conceptual division. In the unweighted case, scalar curvature, volume normalization, and critical Sobolev exponent are rigidly linked by conformal covariance. That rigidity underlies Aubin’s comparison with the sphere, spherical symmetrization, surgery thresholds, local compactness criteria, and the precise asymptotics on product manifolds. In the weighted case, analogous quotients exist, but the comparison geometry changes from $M^n$74 to model constants $M^n$75 on Euclidean space with parameter $M^n$76. A plausible implication is that many of the strongest explicit lower bounds and gap theorems remain easier in the unweighted setting because the model spaces and sharp constants are more directly controlled.

Across the papers considered here, unweighted Yamabe-type constants serve four recurrent functions. First, they measure optimal conformal scalar-curvature geometry within a fixed class. Second, they act as compactness thresholds against concentration, both on smooth and singular spaces. Third, they control surgery and bordism by means of explicit model-space constants. Fourth, their higher, relative, equivariant, spinorial, and CR analogues detect finer geometric phenomena such as nodal solutions, boundary rigidity, symmetry breaking, and sign changes. In that sense, “unweighted Yamabe-type constants” designate not a single invariant but a coherent family of scale-invariant conformal quantities governed by the same unweighted variational principle.