Minimal Surfaces in R 3

Transcription

1 Minimal Suaces in R 3 AJ Vargas May 10, Introduction The goal of this note is to give a brief introduction to the theory of minimal suaces in R 3, and to show how one would go about generalizing the theory to oriented k-dimensional hyersuaces in R n. We will also touch on some interesting roblems, alications, and results related to the field. We will touch on Plateau s laws for soa films, Plateau s roblem, the Double Bubble theorem, and finally show some ictures. First let s say what we mean by a minimal suace. Roughly, a minimal suace is one for which each oint has neighborhood which, as itself a suace, has smallest ossible area among suaces sharing the same boundary. That is, a minimal suace is one which locally minimizes area. To have a concrete idea of how to come u with such things, let s begin with the classical derivation of the minimal suace equation as the Euler-Lagrange equation for the area functional, which is a certain PDE condition due to Lagrange circa 176 describing recisely which functions can have grahs which are minimal suaces. By viewing a function whose grah was a minimal suace as a minimizing function for a certain area functional (i.e. a smooth ma from a function sace into R which measures area ), we are able to use techniques from the calculus of variations to show that for an oriented suace, locally minimizing area is equivalent with the vanishing of the divergence of the unit normal vector to that suace. It is from this ersective that we can generalize the situation to arbitrary k-dimensional submanifolds of R n, and in fact to any k-dimensional submanifold of a Riemannian manifold. Recall that any suace is locally the grah of some smooth function. This is a relatively easy consequence of the inverse function theorem, based on what it means to be a regular arametrized suace, that is, where the two co-ordinate vectors for the arametrization san a two dimensional vector sace. Then by the above discussion, to find more general minimal suaces it will su ce to find minimal grahs of functions. Let 6=? be a bounded oen set in R, and let :! () R 3 be any di eomorhism. We should think of the the set (@) asaclosed wire insace. Inthesettingofaregularsuace,thisrolewill be filled by the boundary of the neighborhood in which the suace can be viewed as the grah of a function. In any case, let s consider the set H of functions h C 1 () suchthat h(@) = (@) (i.e. whoseboundariesarethegivenwire). We llsaythatthegrahof f H has minimal area with resect if 1

2 Z Z d f ale d h for each h H (here h is just the grah h, with the usual arametrization, and d h is the area element det(i) dx ^ dy, I being the first fundamental form of h). Say that f has minimal area if its grah does. In other words, f has minimal area with resect to if it is a minimizer of the area functional a : H! R defined by Z a(h) = d h. Here is how the minimizing functions were characterized. Suose f minimizes a. Then chose any other smooth H so that s value is identically 0. We can get a arametrized family of functions z t = f + t H. The term t is called a variation of f. Since f was chosen to be a minimizing function, t =0mustbeacriticalointforthe function That t =0iscriticalmeansthat It s not hard to check that Z A(t) =a(z t )= d( zt ) Z q = 1+(z t ) x +(z t ) y dx ^ dy = 1+ rzt dxdy. d dt A(t) = d 1+ rzt t=0 dt dxdy t=0 d = 1+ rzt dt dxdy =0. d 1+ rzt dt = = hrz t, r i 1+ rzt t=0 rz t 1+ rzt, r so that after evaluating at t =0,equation(1)becomes d dt A(t) = t=0 = =0. d 1+ rzt dt dxdy t=0, r dxdy (1)

3 Now, alying the formula div('f) =hr', Fi+' div(f) (where ' : R n! R is smooth and F is a smooth vector field) we get div, r = div so that integrating both sides and alying the divergence theorem, we can rewrite equation (1) again as d dt A(t) = t=0 = = = =0, r dxdy div dxdy div dxdy div dxdy Notice that we could conclude div dxdy ds. ds =0because was chosen to vanish We can now aly the fundamental lemma of the calculus of variations, which states that if R n is an oen set, and f is a continuous function on, then if for all comactly suorted functions ' C 1 ( ) onehas Z '(x)f(x)dx, then f is identically 0 on. In our resent situation, any function ' which is comactly suorted on will extend to a function ' C 1 () whichwillvanishon@, so we can conclude, as desired, that f x f y = trace(d) = 0 on all of if and only if f is a minimizer of the area functional a. Equation () is called f the minimal suace equation. Here, D is the Jacobian matrix of the function. If we comute all the artials and simlify, we get the following PDE f xx (1 + f y ) f x f y f xy + f yy (1 + f x)=0. (3) It can be shown that any solution to this equation must be real-analytic. This demonstrates that functions of minimal area are not so readily abundant. As a final note, it 3

4 turns out that the shae oerator of any grah of minimal area arametrized via r(x, y) = (x, y, f(x, y)) has no trace anywhere, because the trace of this ma turns out to be exactly the minimal suace equation (3) u to some non-zero multile. Let s see this. It s not hard to comute the coe cients of the first and second fundamental forms of a function as E =1+fx F = f x f y G =1+fy f e = xx f f = xy f g = yy 1+f x +f y and therefore the mean curvature is 1+f x +f y 1+f x +f y eg ff + ge (EG F ) = f xx(1 + f y ) f x f y f xy + f yy (1 + f x) (1 + f x + f y ) 3 =0. Since the trace of the shae oerator of a suace is indeendent of choice of arametrization, one may instead say a arametrized suace is minimal whenever its shae oerator has no trace anywhere. This definition rovides us with some more geometric ersective. Mean curvature tells us the average value of the normal curvatures of the suace over all ossible directions. It is well known that the mean curvature is determined by the average of the eigenvalues of the shae oerator, which corresond resectively to the maximum and minimum of the rincial curvatures of the suace at a oint over all ossible directions. This directions of maximal and minimal curvature are called rincial directions. So namely, the trace of the shae oerator being 0 everywhere tells us that the rincial curvatures at each oint cancel each other. This means that for all oints on the suace, the curvature of acurveintherincialdirectionofthesuacewillbethenegativeofthecurvatureinthe other rincial direction. Let us now try to generalize the situation. Basically the same rocedure we outlined above will make sense for an arbitrary k-submanifold of R n, but first we have to say clearly what the area (volume more generally) element is for such a thing. For a regular suace, the most natural definition is that the area element would be the size of the area of the arallelogram sanned by the co-ordinate tangent vectors to the suace at a oint times the usual area element dx ^ dy. In general, any k-dimensional linear subsace of some n- dimensional inner roduct sace is itself an inner roduct sace via the induced inner roduct on the bigger sace. We already know from linear algebra that the area of the arallelogram sanned by the linearly indeendent set {v 1,...,v k } of vectors is gotten by taking the size of the determinant of the symmetric matrix which encodes how the inner roduct oerates on the subsace generated by the vectors. The matrix in other words encodes all of the information about lengths and angles in the san of the set. This matrix is called the Gram matrix of the linearly indeendent set. It is given by G(v 1,...,v k )=(v 1... v k ) T (v 1... v k ), or erhas more clearly, [G(v 1,...,v k )] ij = hv i,v j i. All local geometric roerties (i.e. those relating to concets of size (length, area, volume etc) and angle) of a submanifold of R n are described by the way the canonical basis for the tangent sace at a oint inherits an inner roduct structure from R n. In the very articular situation of a regular suace 4

5 S arametrized via r : R! R 3, we have that the vectors r @z form a basis for the tangent sace at the oint = r(u, v). @v Gram matrix is r G(r u,r v )=(r u r v ) T T (r u r v )= u r u ru T r v rv T r u rv T r v hru,r = u i hr u,r v i hr v,r u i hr v,r v i = So that the Gram matrix is exactly the first fundamental form of the suace at a oint. This is clearly the right generalization. We can now define the area q element for an arbitrary immersed k-dimensional submanifold S k of R n as simly ds = det G(r u1,...,r uk ) dv where dv is the usual volume form on R n and r is the immersion, r ui are the basis vectors for the tangent sace. When our submanifold is an oriented hyersuace, we are able to basically reackage the above argument to get that a function minimizing the volume functional v(h) = R d h (where now h is a smooth R n 1! R function) is equivalent with the divergence of the unit normal vector field (which we can define because we re working with an oriented hyersuace) of the grah of that function vanishing. So we can study minimal hyersuaces too. The Helicoid and the Catenoid Let s look closely at an examle of a minimal suace. Suose f is minimal, and consider alevelcurveoff given imlicitly by f(x, y) =c for some constant c. It s not hard to comute the curvature k of this curve as k = f xxf y +f x f y f xy f yy f x 3. Should f have level curves which are straight lines, it must be that k 0, imlying that f is harmonic. To see this, simly notice that we can simlify equation (1) to f xx (1 + f y ) f x f y f xy + f yy (1 + f x)=(f xx + f yy )+(f xx f y f x f y f x + f x) = f xx + f yy =0. It turns out that the only harmonic functions with linear level curves are of the form y y0 f(x, y) = arctan + µ x x 0 for constants,µ,x 0,y 0 R. 5

6 If 6= 0,thegrahofsuchafunctioniscalledahelicoid (otherwise we get a lane, a somewhat trivial minimal suace). To see what s haening, let s work with a more illuminating arametrization. It can be shown that the following arametrization gives the same suace as f. Remark. We are taking the whole lane as our domain here, but there s no need for that. We could instead consider comact helicoids and get the same results. x = u cos v + x 0 y = u sin v + y 0 z = v + µ. Figure 1: A Helicoid This suace is secial in that it is ruled, meaning that it is gotten by sweeing a line along some base curve, admitting a arametrization of the form r(u, v) = b(v)+u (v), where b is an analytic curve (called the directrix or base curve) erendicular to the rulings (i.e. the individual lines) of the suace, and (v) isaunitvectorinthedirectionoftheruling 6

7 through r(u, v). Indeed, notice that we can rewrite the given arametrization r for the helicoid as r(u, v) =(0, 0,v)+u cos v, sin v, 0, so it is ruled, and indeed it can be easily seen as a artition into lines along helixes. Of articular interest is that any ruled minimal suace (excet the lane) can be isometrically embedded into some helicoid. Thus the helicoid actually characterizes all ruled minimal suaces. As an exercise in comuting suace curvatures, let s verify that the helicoid has no mean curvature. In what follows, ut X = r(r ) R 3. Since we ll be working with a single smooth regular arametrization r : R! R 3, we get a natural choice of smooth non-vanishing unit normal vector field Ñ gotten by taking the cross roduct of the basis vectors (r u,r v )int X. Namely, Ñ(u, v) = r u ^ r v kr u ^ r v k. This gives us the outward facing normals. Note Ñ only has the roerties we claim because r is regular (i.e. r u and r v are linearly indeendent). Then the Gauss ma (with resect to the orientation Ñ) is the smooth ma N : X! S gotten by N(u, v) =Ñ(u, v). We have r @z = cos v, sin v, 0 @u @z = u sin v, u @v Thus, so that r u ^ r v = sin v, cos v, u, N(r(u, v)) = r u ^ r v kr u ^ r v k = sin v + u, cos v + u, u + u =(N 1,N,N 3 ). The idea here is to generalize Euler s characterization of curvature as the infinitesimal change of angle between tangent vectors with resect to osition along the curve in order to get a curvature measure. Then the natural generalization for the curvature of a suace at aoint = r(u, v) would be the Jacobian determinant of N (the Jacobian matrix dn of N is called the shae oerator). This number is called the Gaussian curvature of X at. We might as well start by comuting the Gaussian curvature, as we need to coe cients of the fundamental forms in order to comute mean curvature anyway. Recalling that by definition dn (r u )=N u and dn (r v )=N v where N u @N = u sin v ( + u ), u cos v 3 ( + u ), u (1 u + u ) (1+ u + u ) and N v @N = cos v + u, u sin v + u, 0. 7

8 We seek to write N u = a 11 r u + a 1 r v and N v = a 1 r u + a r v, then the Gaussian curvature K will simly be the number a 11 a a 1 a 1. Thankfully we can aeal to the Weingarten equations to comute these a ij. That is, a 11 = eg ff a EG F 1 = a 1 = fe ef a EG F = fg gf EG F ge ff EG F, where (E,F,G)and(e, f, g) arethecoe of X, resectively. Recalling that cients of the first and second fundamental forms E=r u r u F=r u r v G=r v r v e=n u r u f=n u r v g=n v r v, we can easily conclude that E =1,F =0andG = +u, while e =0,f = +u and g = 0. Then also EG F = G, andeg ff =0,fG gf = fg,fe ef = f,ge ff =0, so that finally we can conclude a 11 =0 a 1 = f a 1 = f G a =0, f thus the Gaussian curvature is =. Finally, the eigenvalues of dn (i.e. the G ( +u ) rincial curvatures) are easily seen to be ±, and thus their sum is 0, meaning that +u the helicoid X has no mean curvature anywhere, and hence is a minimal (arametrized) suace as desired. (Notice here that it is crucially imortant that dn is a self-adjoint linear oerator, because that allows us to orthonormally diagonalize dn thanks to the sectral theorem. This allows us to conclude the rincial curvatures agree with the eigenvalues of dn.) As a final note, since once again dn is self-adjoint, we can exress the Gaussian curvature as simly the roduct of the eigenvalues. This tells us immediately that the Gaussian curvature of any minimal suace is going to be negative, meaning that in articular each oint of a minimal suaces is a saddle oint. Now, we ve seen that angles and lengths of tangent vectors on a suace are comletely determined by the first fundamental form. It stands to reason that any regular arametrized suace s(u, v) which shares the same fundamental form with r at a oint must be locally isometric to r. That is, they measure distances and angles exactly the same way. As an ant walked around from oint to oint, it would not be able to distinguish between these two suaces geometrically. 8

9 It turns out that the helicoid ossess a certain arametrization, called a Weierstrass arametrization, which generated a arametrized family of minimal suaces, each locally isometric to the helicoid. One might imagine the family as a system of local isometric deformations of the helicoid into a dual minimal suace, in this case called a catenoid. Secifically, a Weierstrass arametrization has the form Z x k ( ) =Re 0 k(z)dz + c k where 1 = f(1 g ), = if(1+g ),and 3 = fg, where f is analytic, g is meromorhic (analytic excet at a set of isolated oints), and fg is analytic. This arametrizes an associate family of airwise locally isometric minimal suaces by Z x k (, ) =Re e i 0 for [0, ] andc k R. It can be shown that the arametrization k(z)dz + c k x =cos sinh v sin u +sin cosh v cos u y = cos sinh v cos u +sin cosh v sin u z = u cos + v sin. has this form, with (u, v) (, ] R, (, ]. The values at =0, corresond to helicoids, and the values at = ± corresond to these aformentioned catenoids. Figure : A Catenoid. It can be realized as the suace of revolution generated by a catenary curve. 9

10 Figure 3: A icture of the deformation. 3 Soa Films and Plateau s Problem A soa film is the suace you d get from diing a wire in some bubble soa. The Young-Lalace equation = H relates the suace tension and mean curvature H of an inteace between two fluids (like asoafilm)tothedi erence between the caillary ressures on either side of the inteace. This equation hysically catures why soa films should be minimal suaces, just take the surrounding fluid to be air. The same ressure is exerted from either side of the film, so there is no ressure di erential and hence the film arranges itself to minimize area. As an examle, imagine that two soa bubbles have joined together. The inteace along with they meet is an examle of a soa film (where the wire is the boundary along which the bubbles meet), and the comlex they form is called a double bubble. The Double bubble theorem, roved in 00, asserts that the smallest suace enclosing and searating two given volumes is exactly the double bubble described above. Similar to the isoerimetric inequality. 10

11 Soa films were first studied extensively in the early 19th century by the Belgian hysicist Joseh Plateau, who roosed that soa films always exhibited certain henomena when taken in grous. These roerties were called Plateau s Laws, and they are as follows: Soa films are smooth suaces. Soa films have constant mean curvature everywhere. Soa films always meet in threes, at a mutual dihedral angle of 10. The four lines along which soa films meet intersect at a common vertex, forming a tetrahedron in the sense that, taken airwise, the lines meet at tetrahedral angles. ( ) Figure 4: A foam of soa films demonstrating the above henomena It is a very amazing fact that these conditions can in fact be roven mathematically by modeling soa films in a way which reflects a certain area minimizing rincile for suaces. This was done in 1976 by Frederick Almgren and Jean E. Taylor. The rincile they chose 11

12 to model was that a hysical system would remain in a given geometric configuration only if it cannot readily change to a configuration which requires less energy to maintain. Ignoring the e ects of gravitational otential energy on the configuration of the suace, the only other thing that contributes is the comressional energy sulied by the surrounding air (a static fluid), and the suace energy of the susended fluid (soa in our case). Thus, when there is no contribution from the air ressure, the suace energy er unit suace area must be minimized. Among other things, this has the e ect of minimizing the area of the suace along the boundary, i.e. these soa films are minimal suaces. 4 Plateau s Problem Perhas a natural question to question to ask at this oint is whether there is always a minimal suace with a given boundary. After all, our intuition about soa films would suort this being the case. Figure 5: A helicoid is the minimal suace bounded by a helix. We see a soa film taking on this shae. This is known as Plateau s roblem. The answer is in the a rmative, and mathematician Jesse Douglas won a Fields medal for his solution. It was shown by Gauss that any regular arametrized suace r : B(0, 1) C! R n admits a conformal arametrization (i.e. the Jacobian matrix is a scalar times a rotation matrix, so that the arametrization reserves angles). The first ste was to realize that for a conformal arametrization (so that E = G and F =0forthecoe cients of I ), the minimality condition is equivalent with having the comonent functions r i all harmonic. Thus the Plateau roblem may be formulated as follows: 1

13 Problem 1. Given a contour R n, find a regular arametrization r : B(0, 1)! R n which is harmonic (i.e. all co-ordinate functions are harmonic) and conformal on B(0, 1) and for which the restriction 1) is a regular arametrization for. An imortant fact about harmonic functions defined on oen subsets of the comlex lane is the following: suose u : {} C! R is harmonic excet at ( a domain), and u is bounded in some neighborhood of, then u extends to a harmonic ma on all of. Thisiscalledaharmonic extension of u to. Thus Douglas idea was to find a arametrization g of the contour whose harmonic extension was conformal, giving a minimal suace bounded by and thus solving Plateau s roblem. He showed if g 1)! R 3 arametrized and 1) was any homeomorhism solving the following integral equation Z g 0 (t) g 0 ( ) d =0, (t) ( ) that the harmonic extension of g 1 to the uer half lane by means of the Poisson Kernel is the required harmonic and conformal extension. He found this to be theminimizer of a a certain mysterious functional, called his A-functional, whose definitional motivation remains largely unknown. 5 Further Direction IaminterestedtolookmorecloselyathowthetheorygeneralizestogeneralRiemannian manifolds, and in the future am looking to develo a better understand of the methods of Douglas and Taylor to motivate further study of the more generalized theory of minimal submanifolds. A very secial thanks to Professor Sema Salur, without whose direction, rudence, and kindness, this roject would not have haened. 13

14 6 Other Examles Figure 6: This is an examle of an embedded minimal suace with 3 ends (3 ways to reach infinity. ) It is homeomorhic to a trily-unctured torus. 14

15 Figure 7: The Riemann examles are a arametrized family of minimal suaces which are foliated by circles. They characterize all lanar minimal suaces (boundary is a circles between two arallel lanes. 15