Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. (MN-27)

Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. (MN-27)

by Jon T. Pitts

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Mathematical No/ex, 27

Originally published in 1981.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

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ISBN-13: 9780691615004
Publisher: Princeton University Press
Publication date: 07/14/2014
Series: Mathematical Notes , #696
Pages: 338
Product dimensions: 9.10(w) x 6.10(h) x 0.50(d)

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Existence and Regularity of Minimal Surfaces on Riemannian Manifolds


By Jon T. Pitts

PRINCETON UNIVERSITY PRESS

Copyright © 1981 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08290-5



CHAPTER 1

NONTECHNICAL SYSTEMIC DESCRIPTION AND ILLUSTRATIVE EXAMPLES


This chapter contains a reasonably complete but informal explanation of our methods, accomplished largely through examples and counterexamples. The reader may find this systemic summary useful in several ways. First, it provides an overview of the comprehensive theory which follows. The ideas which are the foundation of this monograph unify the structure and may be illustrated. Second, it motivates the theory. The finished product is written in considerable generality, and it is possible to lose sight of the simple natural origins of the concepts. For example, in chapters 3 and 4, we study a special generalized minimal surface (called an almost minimizing varifold), introduced to help solve just our sort of variational problem. As we shall see, simple examples illustrate how easy and natural it is, in our context, to isolate and study almost minimizing varifolds. Third, we may perhaps save the reader some time. In our experience, there are several ideas, of which one is the almost minimizing varifold, for which it seems that one should be able to substitute an apparently simpler notion. We hope to illustrate why some of these "simplifications" may actually introduce greater complexity. Fourth, apart from general considerations, some of these examples have intrinsic value as examples of interesting geometric phenomena.

(NOTE. Although the examples and explanations below are informal and largely nontechnical, some notation has been unavoidable. Where that is the case, we have been consistent with the conventions of chapter 2.)


1.1 THE VARIATIONAL CALCULUS.

Here we outline a program to find a closed minimal surface on a compact riemannian manifold M , and we describe the pitfalls of the program. In these examples, the manifold M is usually two dimensional, and the sought-after minimal surface is one dimensional (closed geodesic), but the methods work as well in any dimension and codimension.

(1) We begin by describing a simple procedure for finding a nontrivial path of closed surfaces on M. For example, let us suppose that M is the surface in figure 1; that is, M is diffeomorphic to S2 in R3, but with a different metric.

Let P : R3 -> R be orthogonal projection onto the y-axis, and assume M is in general position. Then the projection of M onto the y-axis, P[M] , will be a closed subinterval [a,b], and for almost all numbers a ≤ y ≤ b, M [intersection] P-1 [y] is a closed curve on M (fig. 2). Actually, we do not care much about the projection P, but we are quite interested in the map y -> M [intersection] P-1 [y] which takes points y in [a,b] to closed curves on M.

To make generalization easier, let us reparametrize the interval [a,b] so that the domain is the standard interval [0,1] = I (For definiteness, assume the new parametrization is t -> a + t(b - a) , t [member of] I, although this is not essential.) If we consider the points of M lying over a and b to be degenerate curves, then the reparametrization gives us a map φ whose domain is I and whose range is the space of closed curves on M. Specifically, φ(t) is the curve M [intersection] P-1 (a+t(b-a)). This construction is clearly natural, and it has a dynamic quality: the curves φ(t) "sweep out" all of M as t varies from 0 to 1. (See figure 3.)

Before moving on to the next step, we define

L(φ) = sup {length(φ(t)) 0 ≤ t ≤ 1}.

The number L(φ) is the length of the longest curve in the family (φ(t)}t[member of]I, and it is important. In figure 3, for example, L(φ) might well be the length of the curve φ(3/4).

(2) Now with M as in (1), we describe a plausible procedure for finding a closed one dimensional minimal submanifold (closed geodesic) on M. The argument is by maximum-minimum methods.

In constructing the map φ, we have gone from the particular to the general; we need not have been so orderly. The germane properties of φ are these, roughly speaking. We are interested in continuous paths ψ of closed curves on M. The domain of ψ should be the unit interval I; the curves ψ(0) and ψ(1) should be degenerate {points); and ψ should be nontrivial in the sense that the curves ψ(t) should "sweep out" M as t goes from 0 to 1. One such path ψ, which is somewhat different from φ, is shown in figure 4. For use below, we denote by

Π


the family of all such maps ψ having the properties described. As was done in (1), we may assign to each map ψ [member of] Π the number

L(ψ) = sup (length(ψ (t)) : 0 ≤ t ≤ 1}.


One notes for example that L(ψ) < L(φ) in figures 4 and 3; this is the kernel of the idea for finding a minimal surface on M. We define the critical level of Π,

L(Π) = inf {L(ψ) : ψ [member of] Π};

an element ψ0 [member of] Π is a critical map provided

L(ψ0) = L(Π);


and if ψ0 is a critical map, then C [member of] Image(ψ0) is a critical surface provided

area(C) = L(Π).


(We shall use this convenient terminology throughout this chapter.) For example, ψ is a critical map (in fig. 4) and ψ(1/2) is a critical curve (and a closed geodesic) on M. Speaking very generally, the idea is that one of the critical surfaces of acritical map should be a minimal surface.

(3) Here we discuss a rigorous generalization of the ideas in (2). Suppose 1 ≤ k ≤ m = dim(M). According to [AFl] , there is an isomorphism

(*) Hm (M; Z) [equivalent] πm-k(Zk(M); {0})


of the m dimensional homology group of M with coefficients in the integers and the (m-k)-dimensional based homotopy groups of the k dimensional integral cycle groups [Zk(M)] of M (notation as in 2.17). (We remark that the isomorphism in (*) is natural and is constructed by modifying the slicing argument shown in figure 2.) In (2), M is orientable and m = 2 , so

Z [equivalent] H2 (M; Z) [equivalent] π1 (Z1(M); {0}).


Thus the family Π is the image in π1 of a generator of H1 under the isomorphism; each map φ is a representative of Π; and the curves on M are oriented 1-cycles. The number L(φ), φ [member of] Π, is simply

L(φ) = sup image (M • φ),

where M is the mass norm [FHl, 4.1.7]; and

0 < L(Π) = inf {L(φ) : φ [member of] Π},


since Π is nontrivial. Of course all this works equally well in any dimensions. But--to state exactly where we stand at this point--we have described a rigorous general variational construction; we have not yet produced a minimal surface. We illustrate much of what must be done in the following examples.

(4) In (2), each curve φ(t) of the path φ was an imbedded, continuously differentiable, one dimensional submanifold of M. This need not always be so, as illustrated by the following example, due to Almgren [AF2, p. 15-8]. Here M is diffeomorphic to S2, but is metrized as a "three-legged starfish" (figure 5). One critical path ψ is illustrated, and we have graphed M • ψ (length vs. t) below the manifold. All curves ψ(t) except ψ(c) are simple and closed. Curves ψ(t), t > c, have two components. The critical curve ψ(c) is shaped like a figure eight and has a singularity at the point of intersection. While not conclusive (ψ(c) is properly immersed), this example is suggestive. Manifolds are not closed under the natural operations of geometric measure theory, and apparently the critical surfaces our methods produce might have essential singularities. This is some justification for our working with surfaces at least as general as integral currents.

(5) A second difficulty with the program described in (2) is uniqueness. It is really two problems--one technical and the other generic. The technical problem is illustrated in figure 6. Here M is the unit sphere S2

A critical path of closed curves is illustrated. Obviously, the critical level is 2π, and one of the curves at the critical level is the great circle at t = 1/2. Unfortunately, this particular path of curves is not "efficient." Every curve ψ(t) from t = 1/4 to t = 3/4 oscillates enough to have length 2π exactly, but most are not minimal. Thus it is generally not enough to select any critical surface, because it may not be minimal. (We have called this a technical difficulty because it can be eliminated; one can guarantee that every surface at the critical level is minimal (cf. 4.3(2)).)

The generic difficulty appears in the next example. Here M = S2. Let φ be any critical path of 1-cycles over M for which φ(l/2) is a great circle. Let Φ be any continuous map of the interval {t : 1/4 ≤ t ≤ 1/2} onto the special orthogonal group SO(3) such that Φ(1/4) = Φ(1/2). We define a second critical path ψ.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One checks that every closed geodesic (great circle) on M is a critical curve for ψ. The problem here is a surfeit of minimal surfaces. In general dimensions the problem is compounded because many of the critical surfaces may be minimal and yet possess essential singularities. The problem is to refine the theory sufficiently to find one suitable surface among many competitors.

(6) Here we illustrate why varifolds (2.1(18)) have been introduced into the variational calculus. On a torus M in R3, we shall construct a continuous path of 1-cycles, ψ : I -> [??]1 (M) (notation as in 2.1(17)), such that ψ is a critical map, L(ψ) = 4π,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

but ψ(1/3) = 0. Thus ψ(1/3) vanishes, although ψ(1/3) might have been expected to be a critical surface with mass 4π.

Here are the details. We work in R3 identified with R2 x R. We define functions

G : R2 x R -> R2 x R,

G(x,z) = (-x,z), (x,z) [member of] R2 x R,

and

g : R2 -> R3,

g(r,s) = ((2 +cos s)cos r, (2 + cos s)sin r, sin s), for (r,s) [member of] R2. We let M be the torus

M = g[{(r,s) : -π ≤ r ≤ π, 0 ≤ s ≤ 2π}],

which is the surface of revolution of the circle (x - 2)2 + z2 = 1 , (x,0,z) [member of] R3, about the line {0} X R.

Given r [member of] R and 0 ≤ s < ∞, we define the integral current

T(r,s) = g(r,•) #[0,s]

(notation as in 2.1(17) and [FHl, 4.1.7,4.1.8]). T(r,s) is an oriented arc of some generating circle of revolution of M. In particular, T(r,2π) is a homologically area minimizing oriented circle on M with mass 2π. G#T(r,s) = T(r + π,s) and

M(T(r,s}) = M(G#T(r,s)) = s.

Now let 0 ≤ r < ∞ and s [member of] R. We define the integral current

S(r,s) = g(•, s) #[0,r].


S(r,s) is an oriented arc of a circle on M parallel to the plane reflection of R2 x {0}. The current G#S(r,s) is the refelction of S(r,s) through the line {0} x R, and

M(S(r,s)) = M(G#S(r, s)) ≤ 3 r.


Finally we define an auxiliary function f. Let f : R -> R be a continuous function such that

f(0) = f(2π) = 0,

f(s) > 0 , if 0 < s < 2π, and

s + 3f(s) ≤ 2π, if 0 ≤ s ≤ 2π.


We define the map ψ : I -> [??]1 (M) as follows. If 0 ≤ t < 1/3, then let

ψ(t) = T(f(6πt), 6πt) - T(0, 6πt) + S(f(6πt), 0) - S(f(6πt), 6πt);

if 1/3 ≤ t ≤ 2/3, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

if 2/3 < t ≤ 1, then

ψ(t) = G#ψ(1 - t).


One verifies that ψ has the properties described in the first paragraph.

It is geometrically reasonable that the critical surface V at t = 1/4 should have been the circle (g(0,s) : 0 ≤ s ≤ 2π} with multiplicity two. In this example one recovers V as the limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

notwithstanding the fact that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In practice, in chapter 4 we implement the general theory in just this way (cf. (7) below). We use currents to apply the minimum-maximum construction to homotopy maps of cycles, as in (3); the critical surfaces, however, are varifolds constructed as (varifold) limits of integral cycles.

(7) REMARK. In chapter 4, where the variational calculus is rigorously developed, we define and use generalized homotopy relations instead of the conventional homotopies in (*) of (3). This is largely a technical device (although an important one) employed because the discrete "cut and paste" arguments of geometric measure theory are not easy to apply simultaneously to a continuum of surfaces. These generalized homotopy relations also adapt nicely to the type of combinatorial argument indicated in 1.2 below.


1.2 ALMOST MINIMIZING VARIFOLDS.

Almgren [AF2] employed the variational methods discussed in 1.1 to prove that any compact riemannian manifold supports stationary integral varifolds in all dimensions not exceeding the dimension of the manifold. As simple examples show, stationary integral varifolds may certainly have singularities, so a stronger varifold property is required to prove regularity. Here we discuss the ideas behind one such property of varifolds: almost minimizing.

Let M be a compact manifold and let U be an open subset of M. A varifold V is almost minimizing in U if, roughly speaking, V may be approximated arbitrarily closely in U by integral cycles which are themselves almost locally area minimizing in U. We shall see why such surfaces derive naturally from the minimum-maximum construction in 1.1.

In example 1.1(6) we have already seen a simple example of a critical surface which is an almost minimizing varifold. In the notation of that example, we define a sequence of integral cycles:

Tj = ψ(3-1 + 3-1-j), j = 1,2, ... .


Each current Tj is the sum of two oppositely oriented homologically area minimizing circles on the torus M. Since the varifold [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (as varifolds), the surface V is clearly a critical surface which is the limit of a locally minimizing sequence of integral cycles in M. (We remark that the sequence ψ(3-1 - 3-1-j), j = 1, 2, ... also converges to V (as varifolds), but no current in this sequence is locally area minimizing in M. It is thus not reasonable to expect every sequence of cycles converging to a critical varifold to be locally area minimizing.)

For a more subtle example, we reconsider the three-legged starfish and associated critical path as shown in 1.1(4). The critical surface, we recall, is a figure eight, which we shall denote by V. Let T1, T2, ... be a sequence of cycles in the critical path approaching V from the left, as illustrated in figure 7a. Let q be any point on V except the point of intersection, and let U be a small neighborhood of q. We show U enlarged in figure 7b. One notes that none of the cycles Tj is a minimal surface, hence certainly not locally area minimizing in M. On the other hand, the Tj's are almost locally area minimizing in a sense that can be made precise as follows. In figure 7c, we have constructed a new cycle Sj from Tj. Although it is not shown, Sj agrees with Tj outside of U; inside U, however, Sj is locally area minimizing. Sj is constructed from Tj in a geometrically reasonable way. One verifies that there is a continuous function h: I -> [??]1 (M) such that h(0) = Tj, h(l) = Sj, h(t) = Tj outside U for all t [member of] I, and M • h is nonincreasing; i.e., there is a mass-nonincreasing homotopy from Tj to Sj in U. Obviously, M(Sj) ≤ M(Tj) for each j=1,2, ..., and

(**) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].


In general, there could be many choices for the cycle Sj, depending on which homotopy h (having the properties above) one chooses. The important fact is that for this set U and this sequence {Tj}, equation(**) will always hold, no matter how the sequence {Sj} is constructed from {Tj} ,subject to the above conditions. This is what is meant by the informal statement that the sequence {Tj} is almost locally area minimizing in U. Since the sequence {Tj} converges to V (as varifolds), it follows by definition that V is almost minimizing in U. We have, in fact, virtually proved a theorem (cf. 4.13): for every point q in V except the point of intersection, there is a neighborhood U of q in M such that V is almost minimizing in U. The sequence of almost locally area minimizing integral cycles which converges (as varifolds) to V in U can be chosen from the critical path of cycles over M.

(REMARK. In practice, the construction of the cycles Sj from the cycles Tj is defined sequentially (3.1), and not by continuous homotopies (cf. 1.1(7)).)


(Continues...)

Excerpted from Existence and Regularity of Minimal Surfaces on Riemannian Manifolds by Jon T. Pitts. Copyright © 1981 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents

  • FrontMatter, pg. i
  • CONTENTS, pg. iv
  • INTRODUCTION, pg. 1
  • CHAPTER 1. NONTECHNICAL SYSTEMIC DESCRIPTION AND ILLUSTRATIVE EXAMPLES, pg. 12
  • CHAPTER 2. PRELIMINARIES, pg. 48
  • CHAPTER 3. ALMOST MINIMIZING VARIFOLDS, pg. 91
  • CHAPTER 4. EXISTENCE OF ALMOST MINIMIZING VARIFOLDS, pg. 138
  • CHAPTER 5. POINTWISE CURVATURE ESTIMATES FOR STABLE HYPERSURFACES, pg. 178
  • CHAPTER 6. DECOMPOSITION THEOREM FOR STABLE HYPERSURFACES, pg. 224
  • CHAPTER 7. REGULARITY, pg. 288
  • REFERENCES, pg. 327



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