Isoparametric hypersurfaces and the Yamabe equation Guillermo Henry - - PowerPoint PPT Presentation

Isoparametric hypersurfaces and the Yamabe equation

Guillermo Henry

Universidad de Buenos Aires- CONICET

Workshop on the Isoparametric Theory 2-6 June, 2019, BNU, Beijing.

Isoparametric hypersurfaces and the Yamabe equation

This talk is mainly based on a joint work with Jimmy Petean, CIMAT, Guanajuato, GTO, Mexico.

Isoparametric hypersurfaces and the Yamabe equation

Let (Mn, g) be a closed connected Riemannian manifold of dimension n ≥ 3. A function u : M − → I R is a solution of the Yamabe equation if an∆gu + sgu = cupn−1 for some c ∈ I R, where sg is the scalar curvature of (M, g), an = 4(n−1)

(n−2) , and pn = 2n n−2. (∆I

Rnf = − i ∂2f ∂x2

i )

We are interested in both positive solutions and nodal solutions (i.e., sign changing solutions).

Isoparametric hypersurfaces and the Yamabe equation

Let (Mn, g) be a closed Riemannian manifold (n ≥ 3) and let S ⊂ M be an isoparametric hypersurface. Does there exist a solution of the Yamabe equation (positive/ nodal) that is constant along S? We are going to discuss this problem, especially when the manifold is a Riemannian product. Also we are going to show multiplicity results when M = Sn × Sk is a product of spheres.

Isoparametric hypersurfaces and the Yamabe equation

Geometric meaning of the Yamabe equation:

The conformal class of g is [g] := {fg : f ∈ C ∞

>0(M)} ⊆ M(M).

h ∈ [g] has constant scalar curvature c ⇐ ⇒ h = upn−2g with u a h has constant scalar curvature c ⇐ ⇒ positive solution of h has constant scalar curvature c ⇐ ⇒ the Yamabe equation. an∆gu + sgu = cupn−1

Isoparametric hypersurfaces and the Yamabe equation

Constant scalar curvature metrics in [g] ⇐ ⇒ positive solutions of Constant scalar curvature metrics in [g] ⇐ ⇒ the Yamabe equation. A function f ∈ C ∞(M) is the scalar curvature of some Riemannian metric iff there exists g ∈ M(M) and u ∈ C ∞

>0(M) such that

an∆gu + sgu = f upn−1. In that case, the scalar curvature of the metric upn−2g is supn−2g =f

Isoparametric hypersurfaces and the Yamabe equation

Variational point of view

Scalar curvature functional: h ∈ M(M) − → J(h) =

• M sh dvh

Vol(M, h)

n−2 n

. The critical points are the Einstein metrics (J is not bounded). If we restrict J to a conformal class [g], the critical points are the constant scalar curvature metrics in [g]. Y = J|[g] is called the (Yamabe functional) and is bounded from below. The conformal class [g] is parametrized by C ∞

>0(M):

h ∈ [g] ∼ u ∈ C ∞

>0(M) ←

→ h = upn−2g. u − → Y (u) =

• M an∇u2 + sgu2 dvg

u2

pn

.

Isoparametric hypersurfaces and the Yamabe equation

The Yamabe equation an∆gu + sgu = cupn−1 is the Euler-Lagrange equation of u − → Y (u) =

• M an∇u2 + sgu2 dvg

u2

pn

. Positive solutions of the Yamabe equation ⇐ ⇒ positive critical points of Y ⇐ ⇒ metrics of constant scalar curvature in [g].

Isoparametric hypersurfaces and the Yamabe equation

The Yamabe constant is defined by Y (M, [g]) := inf

h∈[g] Y (h) =

inf

f ∈H2

1(M)−{0} Y (f )

Y (M, [g]) ≤ Y (Sn, [gn

0 ]) = Y (gn 0 ) = n(n − 1)vol(Sn)

2 n .

Actually, Y (M, [g]) is a minimum = ⇒ Y has at least one Actually, Y (M, [g]) is a minimum = ⇒ critical point in [g]. Yamabe Problem In any conformal class there is at least a metric of constant scalar curvature (of a given volume).

• H. Yamabe (1960), N. Tr¨

udinger (1968), T. Aubin (1976), R. Schoen (1984)

Isoparametric hypersurfaces and the Yamabe equation

Meaning of the Yamabe constant

The Yamabe constant determines the sign of the scalar curvature in the conformal class.

• Y (M, [g]) > 0 iff there exist h ∈ [g] with sh > 0.
• Y (M, [g]) = 0 iff there exist h ∈ [g] with sh ≡ 0.
• Y (M, [g]) < 0 iff there exist h ∈ [g] with sh < 0.

Isoparametric hypersurfaces and the Yamabe equation

Meaning of the Yamabe constant

The Yamabe constant determines the sign of the scalar curvature in the conformal class.

• Y (M, [g]) > 0 iff there exist h ∈ [g] with sh > 0.
• Y (M, [g]) = 0 iff there exist h ∈ [g] with sh ≡ 0.
• Y (M, [g]) < 0 iff there exist h ∈ [g] with sh < 0.

The Yamabe equation admits a positive solution u an∆gu + sgu = c|u|pn−2u if and only if sign(c) = sign

• Y (M, [g])
• .

Isoparametric hypersurfaces and the Yamabe equation

Multiplicity of solutions of the Yamabe equation.

If Y (M, [g]) ≤ 0 there is essentially only one positive solution of the Yamabe equation. If (M, g) is not conformal to (Sn, gn

0 ) and there is an Einstein

metric in [g], then there is essentially only one positive solution of the Yamabe equation (the Einstein metric) [Obata] If Y (M, [g]) > 0, could exist several positive solutions (of a given volume).

Isoparametric hypersurfaces and the Yamabe equation

An example

Let (Mm, g) and (Nn, h) be two closed Riemannian manifolds of constant positive scalar curvature and unit volume. (M × N,tmg + t−nh) has constant scalar curvature t−msg + tnsh and unit volume. Therefore, Y (tmg + t−nh) = t−msg + tnsh

t→∞ or t→+∞

− → +∞ Fot t big enough (or t small enough), Y (tmg + t−nh) > Y (Sm+n, [gm+n ]) ≥ Y (M × N, [tmg + t−nh]). Then the constant scalar curvature tmg + t−nh does not miminize Y in [tmg + t−nh]. There exists another constant scalar curvature in [tmg + t−nh]!

Isoparametric hypersurfaces and the Yamabe equation

Positive solutions for (Sn, g n

0 ).

The space of positive solutions of the Yamabe equation on the standard sphere is non-compact. Metrics of constant scalar curvature in [gn

0 ] are of the form

cψpn−2

ǫ

gn

0 ,

with ψε(x) =

• ε

ε2 + d2(x, x0) n−2

2 Isoparametric hypersurfaces and the Yamabe equation

Khuri, Marques and Schoen (2009) If (Mn, g) is not conformal to (Sn, gn

0 ) and n = dim M ≤ 24, then

space of positive solutions of the Yamabe equation is compact in the C 2−topology. Brendle (2008), Brendle and Marques (2009) In each dimension n ≥ 25 there exist examples where the space of solutions is not compact.

Isoparametric hypersurfaces and the Yamabe equation

Why study solutions of Yamabe equation in products?

The Yamabe invariant is an invariant of the differential structure defined by Y (M) := sup

[g]∈C

Y (M, [g]) Y (M) > 0 ⇐ ⇒ there exists g ∈ M(M) with sg > 0.

Isoparametric hypersurfaces and the Yamabe equation

Why study solutions of Yamabe equation in products?

The Yamabe invariant is an invariant of the differential structure defined by Y (M) := sup

[g]∈C

Y (M, [g]) Y (M) > 0 ⇐ ⇒ there exists g ∈ M(M) with sg > 0. Y (Sn) = Y (Sn, [gn

0 ]) and few more Y (Sn × S1) = Y (Sn+1)

(Schoen), Y (I H3/Γ) = −6vol(I H3/Γ, g3

h)

2 3 (Anderson), Y (T n) = 0

(Gromov et al.), Y (CI P2) = 12√π (LeBrun), Y (I RP3) = 2− 2

3 Y (S3) (Bray-Neves).

But not for Y (S2 × S2) =?.

Isoparametric hypersurfaces and the Yamabe equation

Kazdan and Warner (1975):

• Y (M) > 0 iff for any f ∈ C ∞(M) there exists h ∈ M(M)

such that sh = f .

• Y (M) = 0 and there exists g0 such that Y (M, [g0]) = 0,

Then f = sg iif f ≡ 0 o f (p0) < 0 for some p0 ∈ M .

• Y (M) < 0 or Y (M) = 0 but the invariant is not realized,

f = sg iif f (p0) < 0 for some p0 ∈ M.

Isoparametric hypersurfaces and the Yamabe equation

Amman, Dahl and Humbert (2013): If N is obtained from Mn by performing surgeries of co-dimension n − k ≥ 3, then Y (N) ≥ min{Y (M), Λn,k} If n − k ≥ 4 or n − k = 3 and n = 4, 5, 6: Λn,k = inf

c∈[0,1] Y (I

Hk+1 × Sn−k−1, [gk+1

h,c

+ gn−k−1 ]) where gk+1

h,c

is the hyperbolic metric of curvature −c2. Y (Sn × I Rk, [gn

0 + gk e ]) = lim T→0 Y (Sn × Sk, [Tgn 0 + gk 0 ])

[Akutagawa, Florit and Petean (2007)]

Isoparametric hypersurfaces and the Yamabe equation

Amman, Dahl and Humbert (2013): If N is obtained from Mn by performing surgeries of co-dimension n − k ≥ 3, then Y (N) ≥ min{Y (M), Λn,k} If n − k ≥ 4 or n − k = 3 and n = 4, 5, 6: Λn,k = inf

c∈[0,1] Y (I

Hk+1 × Sn−k−1, [gk+1

h,c

+ gn−k−1 ]) where gk+1

h,c

is the hyperbolic metric of curvature −c2. Y (Sn−k−1 ×I Rk+1, [g0 +ge]) = lim

T→0 Y (Sn−k−1 ×Sk+1, [Tg0 +g0])

[Akutagawa, Florit and Petean (2009)]

Isoparametric hypersurfaces and the Yamabe equation

Isoparametric hypersurfaces and the Yamabe equation on Sn × Sk

Let consider (Sn × Sk, g(T) = gn

0 + Tgk 0 ) with T > 0. The

Yamabe equation is an+k∆gn

0 +gk 0 u +

• n(n − 1) + 1

T k(k − 1)

• sgn+k

u = cupn+m−1. We can normalized it: ∆gn

0 +gk 0 u + λu = λupn+m−1,

where λ = n(n − 1) + T −1k(k − 1) an+k

• u ≡ 1 is the trivial solution associated with the c.s.c metric

g(T). If T =

• (k − 1)/(n − 1), then u ≡ 1 is the only solution.

Isoparametric hypersurfaces and the Yamabe equation

We are going to consider solutions of the Yamabe eq. of (Sn × Sk, [gn

0 + Tgk 0 ]) that depend only on Sn.

Conjecture Maybe all the minimizers of the Yamabe constant of a closed Riemannian product depends non-trivially on only one variable. Example: (Sn × S1, gn

0 + Tdt2) [Kobayashi (1987)-Schoen (1989)]

Isoparametric hypersurfaces and the Yamabe equation

Remark: The linearization of the Yamabe eq. at u ≡ 1 is ∆gn

0 +Tgk 0 v = (pn+k − 2)λv.

so it is an eingenvalue problem. The spectrum of ∆gn

0 +Tgk 0 are the set of eigenvalues

Spec(∆gn

0 ) + T −1Spec(∆gk 0 ).

So the eigenvalues are νij =

∆gn

• i(n + i − 1) +

∆T−1gk

• j(k + j − 1)

T .

Isoparametric hypersurfaces and the Yamabe equation

∆gn

0 +Tgk 0 v = (pn+k − 2)λv.

For T small (T <

k n−1),

(pn+k − 2)λ = νij =

∆gn

• i(n + i − 1) +

∆T−1gk

• j(k + j − 1)

T with j ≥ 1. So in that case, for an accurate T (pn+k − 2)λ = i(n + i − 1) Eingenfunctions associated to (pn+k − 2)λ depend only on Sn (the big sphere).

Isoparametric hypersurfaces and the Yamabe equation

A positive solution of Yamabe eq. in (Sn × Sk, gn

0 + Tgk 0 ) that

depends only on Sn satisfies ∆gn

0 u + λu = λuq

• Eλ,q
• ,

where q = pn+k − 1. Note that pn+k < pn, so it is a subcritical equation, H2

1(Sn) is

compactly embedded into Lpn+k(Sn) Bidaut Veron and Veron (1991): If λ ≤

n q−1 then u ≡ 1 is the only solution of Eλ,q.

Thus If T > k−1

n , u ≡ 1 is the only solution of Eλ,q

The goal is to find out non-trivial sol. of Eλ,q for T ≤ k − 1 n < k n − 1.

Isoparametric hypersurfaces and the Yamabe equation

Radial solutions

A radial solution ∆gn

0 u + λu = λuq is constant along the level sets

• f the height function f

f : Sn − → I R f (x) = xn+1 S = f −1(a) ∼ Sn−1 if |a| = 1. f is an isoparametric function of degree l = 1 Radial solutions (existence and multiplicity results) were studied by Jin, Li and Xu (2008).

Isoparametric hypersurfaces and the Yamabe equation

Radial solutions

A radial solution ∆gn

0 u + λu = λuq is constant along the level sets

• f the height function f

f : Sn − → I R f (x) = xn+1 S = f −1(a) ∼ Sn−1 if |a| = 1. f is an isoparametric function of degree l = 1 Radial solutions (existence and multiplicity results) were studied by Jin, Li and Xu (2008). Non-radial solutions?

Isoparametric hypersurfaces and the Yamabe equation

More solutions... Let S ⊂ Sn be an isoparametric hypersurface. We will see that there exist solutions of the Yamabe equation on (Sn × Sk, gn

0 + Tgk 0 ) that are constant along S × Sk .

Isoparametric hypersurfaces and the Yamabe equation

More solutions... Let S ⊂ Sn be an isoparametric hypersurface. We will see that there exist solutions of the Yamabe equation on (Sn × Sk, gn

0 + Tgk 0 ) that are constant along S × Sk .

Let f : Sn − → I R be the isoparametric function that induces S. That is, S is a regular set of f and ∇f 2 = b(f ), ∆gn

0 f = a(f ),

for some functions b ∈ C 2(I) and a ∈ C(I). We denote with Sf the space of functions that are constant along the level sets of f .

Isoparametric hypersurfaces and the Yamabe equation

Recall that λr,n = r(r + n − 1) is the rth−eigenvalue of ∆gn

0 ,.

Theorem [H-Petean] Let S be an isoparametric hypersurface of degree l. If λ(q − 1) = n(n − 1) + T −1k(k − 1) n + k − 1 ∈

• λil,n, λ(i+1)l,n
• ,

then there exist i solutions of ∆gn

0 u + λu = λuq=pn+k−1

(actually for any q < pn) that are constant along S (they belong to Sf ).

Isoparametric hypersurfaces and the Yamabe equation

Recall that λr,n = r(r + n − 1) is the rth−eigenvalue of ∆gn

0 ,.

Theorem [H-Petean] Let S be an isoparametric hypersurface of degree l. If λ(q − 1) = n(n − 1) + T −1k(k − 1) n + k − 1 ∈

• λil,n, λ(i+1)l,n
• ,

then there exist i solutions of ∆gn

0 u + λu = λuq=pn+k−1

(actually for any q < pn) that are constant along S (they belong to Sf ). In [g(T) = gn

0 + T −1gk 0 ] there exist metrics with constant scalar

curvature of the form upn+k−2g(T) with u ∈ Sf (at least i!)

Isoparametric hypersurfaces and the Yamabe equation

Bifurcation theory

Let X be a (neighborhood) of a Banach space and let H : X × I R − → X a C r−map such that H(0, µ) = 0 for any µ ∈ I R. We say that (0, µ0) is a bifurcation point is any neighborhood of (0, µ0) contains a non-trivial zero, (x = 0, µ) s.t. H(x, µ) = 0.

Isoparametric hypersurfaces and the Yamabe equation

Bifurcation theory

Let X be a (neighborhood) of a Banach space and let H : X × I R − → X a C r−map such that H(0, µ) = 0 for any µ ∈ I R. We say that (0, µ0) is a bifurcation point is any neighborhood of (0, µ0) contains a non-trivial zero, i.e., (x = 0, µ) s.t. H(x, µ) = 0.

Isoparametric hypersurfaces and the Yamabe equation

Eingenvalues of ∆g n

0 |Sf :

If f is an isoparametric function of (M, g), then ∆g(Sf ) ⊂ Sf . The spectrum of ∆g|Sf is

• 0 < λf

1(∆g) < λf 2(∆g) < · · · < λf k(∆g) −

→ +∞

• ⊂ Spec∆g

dim Eλf

i = dim

• ker
• ∆g|Sf − λf

i (g)

• = 1

Isoparametric hypersurfaces and the Yamabe equation

∆g n

0 (Sf ) spectrum for Sn

For (Sn, gn

0 ) and f : Sn −

→ [−1, 1] an isoparametric function of degree l λf

i (gn 0 ) = il

• il + n − 1
• with associated eigenfunction

vi = pi ◦ f , where pi is monic polynomial with i simple roots in (−1, 1).

Isoparametric hypersurfaces and the Yamabe equation

The strategy to prove the theorem is to find out some appropriate map F such that for any (u, λ) ∈ Sf × I R F(u, λ) = 0 ← → u is sol. Yamabe eq. with constant λ Study the bifurcation points of F and how non-trivial solutions behave around them.

Isoparametric hypersurfaces and the Yamabe equation

Taking u = v + 1 (q = pn+k − 1) the map F : C 2,α ∩ Sf × I R − → C 2,α(M) F(v, λ) = ∆gn

0 (v + 1) + λ

• (v + 1) − (v + 1)q

F(v, λ) = 0 iff u is a solution of Yamabe equation with constant λ. So F(0, λ) = 0. The linearization at (0, λ0) of F is Fv(0, λ)(w) = ∆gn

0 w + λ0(1 − q)w

F is a Fredholm map. If (0, λ0) is a bifurcation point then λ0(q − 1) is an eingenvalue of ∆gn

0 |Sf :

λ0 = il(il + n − 1) q − 1 = il(il + n − 1) pn+k − 2

Isoparametric hypersurfaces and the Yamabe equation

Taking u = v + 1 (q = pn+k − 1) the map F : C 2,α ∩ Sf × I R − → C 2,α(M) F(v, λ) = ∆gn

0 (v + 1) + λ

• (v + 1) − (v + 1)q

F(v, λ) = 0 iff u is a solution of Yamabe equation with constant λ. So F(0, λ) = 0. The linearization at (0, λ0) of F is Fv(0, λ)(w) = ∆gn

0 w + λ0(1 − q)w

F is a Fredholm map. If (0, λ0) is a bifurcation point, then λ0(q − 1) is an eingenvalue of ∆gn

0 |Sf :

λ0 = il(il + n − 1) q − 1 = il(il + n − 1) pn+k − 2

Isoparametric hypersurfaces and the Yamabe equation

Local bifurcation- Bifurcation points

Using the local bifurcation theorem of Crandall-Rabinowitz we can see that these are all the bifurcations points.

Isoparametric hypersurfaces and the Yamabe equation

Global Bifurcation, Non- trivial solutions and multiplicity.

Let A :=

• (v, µ) ∈ C 2,α(M) ∩ Sf × I

R : v > −1, µ > 1

• .

Let T = (∆gn

0 + Id)−1

For µ = (q − 1)λ + 1 let G : A − → C 2,α(Sf ) defined by G(v, µ) = λT((v + 1)q − qv − 1). Let H(v, µ) = v − µT(v) − G(v, µ). H(v, µ) = 0 ⇐ ⇒ ∆gn

0 u + λu = λuq Isoparametric hypersurfaces and the Yamabe equation

u = v + 1 and µ = λ(q − 1) + 1 H(u, µ) = v − µT(v) − G(v, µ). Since T is compact, G(v, µ) is uniformly bounded in a region where µ is bounded, then we can apply Krasnoselski’s Theorem. Krasnoselski’s Theorem (0, µ) is a bifurcation point ⇐ ⇒ 1

µ is an eingenvalue of T with

• dd multiplicity.

Isoparametric hypersurfaces and the Yamabe equation

u = v + 1 and µ = λ(q − 1) + 1 H(u, µ) = v − µT(v) − G(v, µ). Since T is compact, G(v, µ) is uniformly bounded in a region where µ is bounded, then we can apply Krasnoselski’s Theorem. Krasnoselski’s Theorem (0, µ) is a bifurcation point ⇐ ⇒ 1

µ is an eingenvalue of T with

• dd multiplicity.

1 µ is an eingenvalue of T ⇐ ⇒ µ − 1 is an eingenvalue of ∆gn

0 |Sf

(0, µ) is a bifurcation point ← → µ − 1 = λ(q − 1) = il(il + n − 1)

Isoparametric hypersurfaces and the Yamabe equation

Isoparametric hypersurfaces and the Yamabe equation

Types of branches that are not possible

Let Ci be the connected component of {(v, µ) : H(v, µ) = 0 and v = 0} that contains the bifurcation point (0, µi = 1 + λil).

Isoparametric hypersurfaces and the Yamabe equation

u = v + 1 ∈ Ci non-trivial, u = ϕ ◦ f , f of degree l. Using the M¨ unzner-Cartan polynomial of f we know a(t) and b(t). Then ϕ satisfies de non-linear second order ODE: − l2(1 − t2)

• b(t)

ϕ′′(t) +

• l(l + n − 1)t + c

2l2

• a(t)

ϕ′(t) + λϕ = λϕpn+k−1

Isoparametric hypersurfaces and the Yamabe equation

The non trivial solutions in Ci have i − 1 peaks

By Crandall-Rabinowitz’ s local bifurcation theorem, non-trivial solution close to (0, µi) are parametrized (for small s) by vs = spi + o(s2) So vs has i−zeroes − → any v ∈ Ci has i − zeroes (i − 1 extrema) Thus (0, µj) / ∈ Ci

Isoparametric hypersurfaces and the Yamabe equation

The non trivial solutions in Ci have i − 1 peaks

By Crandall-Rabinowitz’s local bifurcation theorem, non-trivial solution close to (0, ui) are parametrized (for small s) by vs = spi + o(s2) so vs has i−zeroes − → any v ∈ Ci has i − zeroes (i − 1 extrema) Thus (0, µj) / ∈ Ci

Isoparametric hypersurfaces and the Yamabe equation

We can use the Rabinowitz’s Theorem to prove that all branches Ci are compact. But they can not be uniformly bounded in the parameter µ.

Isoparametric hypersurfaces and the Yamabe equation

v = u + 1, µ = λ(q − 1) + 1. H(v, µ) = 0 iff u is a solution of the Yamabe eq. with parameter λ

Isoparametric hypersurfaces and the Yamabe equation

v = u + 1, µ = λ(q − 1) + 1. H(v, µ) = 0 iff u is a solution of the Yamabe eq. with parameter λ

Isoparametric hypersurfaces and the Yamabe equation

v = u + 1, µ = λ(q − 1) + 1. H(v, µ) = 0 iff u is a solution of the Yamabe eq. with parameter λ

Isoparametric hypersurfaces and the Yamabe equation

λ(q − 1) ∈

• λil, λ(i+1)l
• then we have i−solutions of the Yamabe

equation. In terms of the scalar curvature of (Sn × Sk, g(T)) sg(T) = n(n − 1) + T −1k(k − 1) ∈ (n + k − 1)

• λil, λ(i+1)l
• Isoparametric hypersurfaces and the Yamabe equation

λ(q − 1) ∈

• λil, λ(i+1)l
• then we have i−solutions of the Yamabe

equation. In terms of the scalar curvature of (Sn × Sk, g(T)) sg(T) = n(n − 1) + T −1k(k − 1) ∈ (n + k − 1)

• λil, λ(i+1)l
• Let Nn,k(T) the number of isometrically distinct unit volume

metrics with constant scalar curvature in the conformal class [g(T)]. T − → 0 = ⇒ Nn,k(T) − → +∞

Isoparametric hypersurfaces and the Yamabe equation

Using the classification of Isoparametric Hypersurfaces

Corollary Let Ti =

k(k−1) (n+k−1)λi−n(n−1). If T ∈ [Ti+1, Ti) then we have that

a) If n = 2m with m = 2 then Nn,k(T) ≥ i + [(2m − 1)/2][i/2]. b) If n = 2m + 1 with m = 1, 3, 4, 6, 7, 9, 12 then Nn,k(T) ≥ i + m[i/2] + (γ(m) + β(m))[i/4]. c)

• If n = 3, Nn,k(T) ≥ i + [i/2].
• If n = 4, Nn,k(T) ≥ i + [i/2] + [i/3].
• If n = 7, Nn,k(T) ≥ i + 3[i/2] + [i/3] + [i/4] + [i/6].
• If n = 9, Nn,k(T) ≥ i + 4[i/2] + 2[i/4].
• If n = 13, Nn,k(T) ≥ i + 6[i/2] + [i/3] + [i/4] + [i/6].
• If n = 15, Nn,k(T) ≥ i + 7[i/2] + 4[i/4].
• If n = 19, Nn,k(T) ≥ i + 9[i/2] + 3[i/4].
• If n = 25, Nn,k(T) ≥ i + 12[i/2] + [i/3] + [i/4].

Isoparametric hypersurfaces and the Yamabe equation

Positive solutions for general Isoparametric Hypersurfaces

Theorem[H-Petean] Let (Mn, g) and (Nk, h) be two closed Riemannian manifold of constant scalar curvature. Let f be an isoparametric function of (M, g). If sg + sh > an+kλf

1(∆g)

pn+k − 2 then there exist a non-constant function u ∈ Sf that is a solution

• f the Yamabe equation.

Isoparametric hypersurfaces and the Yamabe equation

Nodal solutions of the Yamabe equation and isoparametric hypersurfaces

Let (Mn, g) be a closed connected Rieamannian manifold (n ≥ 3). Nodal solution of the Yamabe equation are sign changing solution

• f

an∆gu + sgu = c|u|pn−2u Let f be an isoparametric function of (M, g), such that sg is constant along the level set of f (sg ∈ Sf )

Isoparametric hypersurfaces and the Yamabe equation

Let d(t) := min

• dimension of the connected components of f −1(t)
• .

If we assume that k(f ) := mint{d(t)} ≥ 1 we have that Theorem[H] There exists a nodal solution of the Yamabe equation that is constant along the level sets of f .

Isoparametric hypersurfaces and the Yamabe equation

Let d(t) := min

• dimension of the connected components of f −1(t)
• .

If we assume that k(f ) := mint{d(t)} ≥ 1 we have that Theorem[H] There exists a nodal solution of the Yamabe equation that is constant along the level sets of f .

• Compact embedding of H2

1(M) ∩ Sf into Lpn when the dimension

• f the level sets are positive.
• Using of the second Yamabe constant.

Results on the existence of nodal solutions induced by isoparametric functions on Riemannian products.

Isoparametric hypersurfaces and the Yamabe equation

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Isoparametric hypersurfaces and the Yamabe equation