Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc

Nikolaos Kapouleas; Martin Man-chun Li

doi:10.1515/crelle-2020-0050

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Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc

Nikolaos Kapouleas und Martin Man-chun Li

Veröffentlicht/Copyright: 24. Februar 2021

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2021 Heft 776

Abstract

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disc. The surfaces are constructed by singular perturbation methods and have three boundary components. They are the free boundary analogue of the Costa–Hoffman–Meeks surfaces and the surfaces constructed by Kapouleas by desingularizing coaxial catenoids and planes. It is plausible that the minimal surfaces we constructed here are the same as the ones obtained recently by Ketover by using the min-max method.

Funding statement: Martin Li would like to thank the Croucher Foundation for the financial support and the Department of Mathematics at Massachusetts Institute of Technology, where part of the work in this paper was done. He was partially supported by CUHK Direct Grant for Research C001-4053118 and a grant from the Research Grants Council of the Hong Kong SAR, China (Project No. CUHK 24305115). Nikolaos Kapouleas was partially supported by NSF grants DMS-1105371 and DMS-1405537.

A Local exponential map estimates

Proposition A.1.

Let g be a Riemannian metric on Bn with coordinates x1,…,xn. Let gi⁢j:=g⁢(∂xi,∂xj) be the metric components in this coordinate system, let gi⁢j be the inverse and let Γi⁢jk be the Christoffel symbols. Suppose that

(A.2)∥gi⁢j:C4(Bn,g0)∥≤c1 𝑎𝑛𝑑 c1-1g0≤g

for some constant c1>1. Then there exists a constant C depending on c1 (and n) such that

(A.3)∥gi⁢j:C4(Bn,g0)∥≤C,∥Γi⁢jk:C3(Bn,g0)∥≤C,

and that the exponential map exp:B1-2⁢C-1n×BC-1n→Bn with respect to g is a well-defined C3-map such that for any multi-indices I, J with |I|+|J|≤3, we have the pointwise estimates

(A.4)|∂xI|I|⁡∂vJ|J|⁡(exp⁡(x,v)-x-v)|≤C⁢|v|max⁡(2-|J|,0),

where |⋅| denotes the norm of a vector with respect to the Euclidean metric g0.

Proof.

The inverse of a matrix A is given by A-1=(det⁡A)-1⁢adj⁡(A), where adj⁡(A) is the adjoint matrix of A. From (A.2) we get estimate (A.3) and that the metrics g∼c1g0 are uniformly equivalent (recall Definition 1.11). From the definition of exponential map

exp⁡(x,v):=γx,v⁢(1),

where γx,v⁢(t):[0,1]→Bn is the unique geodesic (relative to g) starting at x with initial velocity v, that is γx,v⁢(0)=x and γx,v′⁢(0)=v. In other words, γx,v is the unique solution to the geodesic equation with such initial conditions (here γ=(γ1,…,γn) are the coordinate expression of γ)

(A.5){(γk)′′⁢(t)=Γi⁢jk⁢(γ⁢(t))⁢(γi)′⁢(t)⁢(γj)′⁢(t),k=1,2,…,n,γ⁢(0)=x,γ′⁢(0)=v.

By standard ODE theory and the estimates in (A.3), the exponential map exp is well defined for (x,v)∈B1-2⁢C-1n×BC-1n for some constant C depending on c1.

It remains to prove (A.4). The smoothness of the exponential map is a direct consequence of the smooth dependence on initial conditions (x,v) for the solutions to the ODE system (A.5). We will show how to get C1-bounds here. The proof for higher derivatives are similar.

Let |⋅| and ∥⋅∥ be the norm of a vector with respect to g0 and g, respectively. Since γ is a geodesic, it follows that

∥γ′⁢(t)∥≡∥γ′⁢(0)∥=∥v∥.

Using (A.2) and g0∼c1g, we have

|γ′⁢(t)|≤C⁢|v|.

By Taylor’s theorem, g0∼c1g, (A.3) and (A.5), we have the C0-estimate

|exp⁡(x,v)-x-v|=|γ⁢(1)-γ⁢(0)-γ′⁢(0)|

≤maxt∈[0,1]⁡|12⁢γ′′⁢(t)|

≤C⁢maxt∈[0,1]⁡|γ′⁢(t)|2

≤C⁢|v|2.

For estimates on the derivatives, we differentiate the system (A.5). For example, differentiating with respect to some xa:

(A.6){(∂xa⁡γk)′′=(γi)′⁢(γj)′⁢(∇⁡Γi⁢jk⋅∂xa⁡γ)+2⁢Γi⁢jk⁢(γi)′⁢(∂xa⁡γj)′,∂xa⁡γk⁢(0)=δak,∂xa(γk)′(0)=0.

Recall Kato’s inequality that |α⁢(t)|′≤|α′⁢(t)| for any curve α⁢(t) in ℝn. Using (A.6) and (A.3), we have

|∂xa⁡γ′|′≤|∂xa⁡γ′′|≤C⁢|v|2⁢|∂xa⁡γ⁢|+C|⁢v|⁢|∂xa⁡γ′|.

If we define the function G:[0,1]→ℝ by

G⁢(t):=maxs∈[0,t]⁡|∂xa⁡γ′⁢(s)|,

then G is a nonnegative monotone increasing function hence differentiable a.e. and from (A.6), we have the differential inequality

G′⁢(t)≤C⁢|v|2⁢(1+G⁢(t))+C⁢|v|⁢G⁢(t)≤C⁢|v|2+C⁢|v|⁢G⁢(t)

with G⁢(0)=0. Integrating the differential inequality gives

G⁢(t)≤|v|⁢(eC⁢|v|⁢t-1)≤C⁢|v|2,

provided |v| is sufficiently small (but depending only on c1). From this we have the pointwise estimate

|∂xa⁡(exp⁡(x,v)-x-v)|≤C⁢|v|2.

The estimate on the derivatives with respect to v can be obtained similarly. We differentiate (A.5) with respect to some va

(A.7){(∂va⁡γk)′′=(γi)′⁢(γj)′⁢(∇⁡Γi⁢jk⋅∂va⁡γ)+2⁢Γi⁢jk⁢(γi)′⁢(∂va⁡γj)′,∂va⁡γk⁢(0)=0,∂va(γk)′(0)=δak.

|∂va⁡(exp⁡(x,v)-x-v)|≤C⁢|v|.

Higher-order derivative estimates can be obtained in a similar manner. ∎

As a corollary of the above exponential map estimates, one can prove the following lower bound on the injectivity radius.

Corollary A.8.

Under the same assumption as in Proposition A.1, we have

injx⁡(Bn,g)≥C-1

for all x∈B1-2⁢C-1n.

Proof.

Since Dv⁢exp⁡(x,0)=id for all x, using the estimates in Proposition A.1, we have that Dv⁢exp⁡(x,v) is a non-singular matrix for all |v|≤C-1. From this the assertion follows since we are looking at a local coordinate patch. ∎

Acknowledgements

The authors would like to thank Richard Schoen for his continuous support and interest in the results of this article. They would like to thank also the referee for carefully reading the manuscript and making many helpful suggestions.

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Received: 2020-02-24

Revised: 2020-11-19

Published Online: 2021-02-24

Published in Print: 2021-07-01

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