On the fundamental gap and convex sets in

hyperbolic space

Alina Stancu

Asymptotic Geometric Analysis Seminar

May 2, 2020

History of the Problem

• With T. Bourni, J. Clutterbuck, H. Nguyen, G. Wei, V. Wheeler.

• Introduction:

Given Ω ⊂ Rn a domain (connected, compact, w. piecewise

C∞ boundary), the spectrum of the Laplacian −∆ (or, moregeneral, −∆+ V ) with Dirichlet boundary condition

∆u + λu = 0, on Ω

u = 0 on ∂Ω,

is discrete

0 < λ1 < λ2 ≤ λ3 ≤ . . . < λn → +∞.

The fundamental gap of Ω is Γ(Ω) = λ2 − λ1.

Ω can be a domain in any Riemannian manifold M.

History of the Problem

• Upper Bound on the Fundamental Gap:

Payne-Polya-Weinberger (PPW) Conjecture for (Mn,K ), then-dimensional simply connected space of constant curvature K :

For any bounded domain Ω ⊂ (Mn,K ) (note that one does notneed a convexity condition here),

Γ(Ω) := λ2(Ω)− λ1(Ω) ≤ λ2(Bλ1)− λ1(Bλ1

),

where Bλ1is a ball in (Mn,K ) such that λ1(Bλ1

) = λ1(Ω).

The PPW conjecture was proved by Ashbaugh-Benguria (’92) forthe case when K = 0, by Ashbaugh-Benguria (’01) for the casewhen K = 1 (and Ω in the upper hemisphere), and byBenguria-Linde (’07) for the case of K = −1.

Rayleigh Quotient of a Function w on Ω

λ1 = minY1

∫

Ω|∇w |2dx∫

Ωw2dx

and λn = minYn

∫

Ω|∇w |2dx∫

Ωw2dx

,

Y1 = w ∈ C2(Ω) | w 6≡ 0 on Ω,w = 0 on ∂Ω

Yn = w ∈ C2(Ω) | w 6≡ 0 on Ω,w = 0 on ∂Ω, < w , vi >= 0where v1, . . . , vn−1 are the first (n − 1) eigenfunctions.

Idea for proving PPW: Use a smooth function P 6≡ 0 with∫

ΩPv2

1 dx = 0 and

λ2 − λ1 ≤∫

Ω|∇P |2v2

1 dx∫

ΩP2v2

1 dx.

History of the Problem in Rn

• Lower Bound on the Fundamental Gap:

For bounded convex domain Ω in Rn and Schrödinger operator

−∆+V , where V ≥ 0, convex, the question carries a special name:

The Fundamental Gap Conjecture (van der Berg ’83, Yau ’86,Ashbaugh-Benguria ’89)

Γ(Ω,V ) := λ2(Ω,V )−λ1(Ω,V ) ≥ 3π2

D2, where D := Diam (Ω).

The lower bound is approached when V = 0, and the domainis a degenerating thin rectangular box.

On the Fundamental Gap Conjecture in Rn

Progress was slow for a while even in the 1-dimensional case,see early survey by Ashbaugh at AIM, May 2006.

Singer-Wong-Yau-Yau established the bound Γ(Ω,V ) ≥ π2

4D2

using the so-called Li-Yau gradient method, ’85;

later improved by Yu-Zhong, and Ling, ’86 to Γ(Ω,V ) ≥ π2

D2.

NOTE: Key to the SWYY breakthrough was log-concavity ofthe groundstate of Schrödinger operators with convexpotential, [Ashbaugh], earlier work of Brascamp and Lieb.

Further improvement by Yau ’03 depending on upper boundestimate on the log-concavity of the first eigenfunction.

The Fundamental Gap Conjecture is solved with new ideassuch as the modulus of continuity and concavity byAndrews-Clutterbuck ’11. Proofs w/o heat equation followed.

Fundamental Gap on Sn

In 2018, 2019, Dai, He, Seto, Wang, and Wei (in varioussubsets) generalized the fundamental gap of estimate of −∆with Dirichlet boundary conditions to convex domains in S

n,showing that

Γ(Ω) := λ2(Ω)− λ1(Ω) ≥3π2

D2,

with same degenerating case giving the bound.

It is natural to ask if this holds also for convex domains in Hn.

Similarities and Differences

In both settings, the log-concavity of the first eigenfunctionplays an important role.

Just log-concavity is sufficient to obtain λ2 − λ1 ≥ π2

D2 , but toobtain the optimal estimates it is shown that the firsteigenfunction is super log-concave, i.e. more log-concave thanthe first eigenfunction of the one-dimensional model operator,

Ln,K ,D(φ) = φ′′ − (n − 1)tnK (s)φ′ (1)

on [−D

2, D

2] with Dirichlet boundary condition. Here

tnK (s) =

√K tan(

√Ks), K > 0

0, K = 0

−√−K tanh(

√−Ks) K < 0

where K = 0 is the model for Rn & K = 1 is the model for Sn.

K = −1 is not a good model for Hn, though the firsteigenfunction of (1) when K = −1 is still log-concave.

Main Results

Theorem (BCNSWW)

There are convex domains in H2 such that

λ2 − λ1 < 3π2/D2,

where D is the diameter of the domain.

Theorem (BCNSWW)

For any n ≥ 2, any ǫ > 0, and any D > 0 fixed real number, there

exist convex domains in Hn such that

λ2 − λ1 < ǫπ2/D2,

where D is the diameter of the domain.

On the Convex Domains

Let H2 be the hyperbolic space modelled by the Poincaré half-plane(x , y) | y > 0 = (r , θ) | r > 0, θ ∈ (0, π) with the metric

g = ds2 =dx2 + dy2

y2=

dr2

r2 sin2 θ+

dθ2

sin2 θ.

In the orthonormal frame e1 = r sin θ∂

∂r, e2 = sin θ

∂

∂θ, the

non-vanishing Christoffel symbols are Γ211 = −Γ1

12 = −Γ121 = cos θ,

so the Laplace operator becomes

∆v = v11 + v22 = r2 sin2 θ vrr + sin2 θ vθθ + r sin2 θ vr .

On the Convex Domains

We consider the family of domains

Ωc,θ0,θ1= (r , θ) | 1 < r < eπ/c , θ0 < θ < θ1.

θ0 θ1

Ωc,θ0,θ1

P

Q

R

S

(0, 1)

(0, eπ/c )

More on the domains

Why this type of domains?

The following inequalities holds for the diameter D := Dc,θ0,θ1:

arccosh(

csc θ∗ coshπ

c

)

≤ D ≤ arccosh(

csc2 θ∗ coshπ

c+ cot2 θ∗

)

,

where θ∗ := minθ0, π − θ1.

Right inequality holds because our domain is included inΩc,θ∗,π−θ∗ and r.h.s. is its diameter.Left inequality reflects that the diameter is greater than thedistance from P (or R) to the point (0, eπ/c ).

The following limit holds: π2

c2D2

c,θ0,θ1

→ 1 as c → 0 or θ∗ → π2.

The Eigenvalues

Setting u(r , θ) = f (r) h(θ), ∆u = −λu gives

r2 sin2 θ frr h+ r sin2 θ fr h + sin2 θ f hθθ = −λfh,(

r2 frr

f+ r

fr

f

)

+

(

hθθ

h+ λ csc2 θ

)

= 0.

Hence we should solve the two eigenvalue equations

r2frr + rfr = −µf , r ∈ [1, eπ/c ], (2)

hθθ + λ csc2 θ h = µh, θ ∈ [θ0, θ1], (3)

both with Dirichlet conditions. Let t = log r , so (2) becomes

ftt = −µf , t ∈ [0, πc]. (4)

In order for this to satisfy the boundary conditions, µ must bepositive, so we set µ = (kc)2, f (t) = sin(kct), where k ∈ Z

∗.

The First Eigenvalue

The first Dirichlet eigenvalue λ1 of ∆u = −λu on Ωc,θ0,θ1

corresponds to a strictly positive eigenfunction, which impliesthat f in (4) is sin(ct), and so µ = c2 and λ1 is given by thevalue λ solving

hθθ + λ csc2 θ h = c2h, θ ∈ [θ0, θ1],

h(θ0) = h(θ1) = 0,(5)

for h > 0. We denote by λ1 = λc2

1 the smallest such λ.

The Second Eigenvalue

The second eigenvalue λ2 corresponds to a sign-changingeigenfunction: either f or h changes sign. If f changes sign,then f in (4) is given by sin(2ct) and µ = 4c2; in this case λ2

is given by λ4c2

1 solving

hθθ + λ csc2 θ h = 4c2h, θ ∈ [θ0, θ1], (6)

with h > 0 and Dirichlet boundary conditions. Otherwise h

changes sign, f is positive and is given by sin(ct) with µ = c2;then λ2 is given by λc2

2 solving (5) with h changing signexactly once.

Thus, the second eigenvalue is λ2 = minλ4c2

1 , λc2

2 .

Estimating the Eigenvalues

The first eigenvalue of (3): hθθ + λ csc2 θ h = µh denoted byλµ

1 , satisfies

sin2(θ∗)

(

µ+π2

(θ1 − θ0)2

)

≤ λµ1 ≤ µ+

π2

(θ1 − θ0)2. (7)

Proof: Let h be a solution of (3). We multiply both sides ofthe equation by h and integrate, to obtain

λ =

∫ θ1

θ0(|hθ|2 + µh2)dθ∫ θ1

θ0(csc2 θ)h2dθ

>1

csc2(θ∗)

(

µ+

∫ θ1

θ0|hθ|2dθ

∫ θ1

θ0h2dθ

)

and use Wirtinger’s inequality∫

D

0(h′)2 dx ≥ (π/D)2

∫

D

0h2 dx .

From above, choose the test function ϕ = sin(

θ−θ0

θ1−θ0π)

in

Rayleigh quotient. But csc2 θ ≥ 1, so λµ1 ≤ µ+

π2

(θ1 − θ0)2.

Then the following estimate for λµ2 , the second eigenvalue of (3)

holds:

sin2 θ∗

(

µ+4π2

(θ1 − θ0)2

)

≤ λµ2 ≤ µ+

4π2

(θ1 − θ0)2(8)

Combining (8) and (7), we have λc2

2 ≥ λ4c2

1 when

π2

(θ1 − θ0)2(4 sin2 θ∗ − 1)

(4 − sin2 θ∗)≥ c2, θ∗ >

π6. (9)

LemmaAssume that c , θ∗ satisfies (9). Then the fundamental gap of

Ωc,θ0,θ1satisfies

3 sin2 θ∗c2 < λ2 − λ1 < 3c2. (10)

Hence, as θ∗ approaches π2, the gap approaches 3c2.

Application of Sturm’s Comparison Theorem

In fact,

sin2 θ∗[1 +c

πln(csc θ∗)]

2 < (λ2 − λ1)D

2

c,θ0,θ1

3π2 <[

1 +c

π(2 ln(csc θ∗) + η)

]2

,

where η := η(c , θ∗) > 0 and goes to zero if θ∗ ր π/2 or c ց 0.

The left hand side is ≥ 1 when c > π and sin θ∗ ≥ exp(πc− 1). For

any fixed c > π, and for all θ∗ sufficiently close to π2, we have that

c , θ∗ satisfy sin θ∗ ≥ exp(πc− 1) and condition (9).

For these domains, the gap satisfies the inequality λ2 − λ1 > 3π2

D2 .

Consider a family of problems generalizing (3), indexed by aparameter t

h′′ + v(t)h = µ(t)h on I = [θ0, θ1], (11)

with vanishing Dirichlet boundary conditions. Hereh(θ) = ht(θ) depends on t, and v as well, v(t) = λ(t) csc2 θ.Let λ(t) be the first eigenvalue for each t, which is smooth int, and ht(θ) are all first eigenfunctions, so ht(θ) > 0 on(θ0, θ1).

Denoting derivatives with respect to t as h, we get

h′′ + vh + v h = µh + µh on I . (12)

To relate changes in µ with changes in v , we multiply (12) byh, integrate over I , and use (11) to find

µ

∫

h2dθ =

∫

vh2dθ.

Therefore, if we set µ(t) = c2 + 3c2t, we have

3c2

∫

h2dθ =

∫

λ(csc2 θ)h2dθ = λ

∫

(csc2 θ)h2dθ.

Rearrange

λ = 3c2

∫

(ht)2dθ∫

(csc2 θ)(ht)2dθ,

and integrate from t = 0 to 1, recalling λ(0) = λ1, λ(1) = λ2:

Proposition

λ2 − λ1 ≤ 3c2 maxt

∫

(ht)2dθ∫

(csc2 θ)(ht)2dθ≤ 3c2(1 − δ), (13)

for some δ = δ(θ0, θ1) > 0, independent of c .

Combining with the estimate on diameter, we get that for all csufficiently small, and θ∗ >

π6, the fundamental gap of the

domains Ωc,θ0,θ1is less than 3π2/D2.

Conclusions

At the talk (-: