A proof by foliation that Lawson's cones are A_Φ-minimizing
AA PROOF BY FOLIATION THAT LAWSON’S CONES ARE A Φ -MINIMIZING CONNOR MOONEY AND YANG YANG
Abstract.
We give a proof by foliation that the cones over S k × S l minimizeparametric elliptic functionals for each k, l ≥
1. We also analyze the behaviorat infinity of the leaves in the foliations. This analysis motivates conjecturesrelated to the existence and growth rates of nonlinear entire solutions to equa-tions of minimal surface type that arise in the study of such functionals. Introduction
A well-known result in the theory of minimal surfaces is that area-minimizinghypersurfaces in R n +1 are smooth when n ≤
6, but can have singularities in higherdimensions. An important tool in the theory is the monotonicity formula, whichreduces the regularity problem to establishing the existence or non-existence ofsingular area-minimizing hypercones. Such cones were ruled out by Simons in thecase n ≤ S × S in R is area-minimizing [3].In this paper we consider the regularity problem for minimizers of parametricelliptic functionals, which generalize the area functional. These assign to an orientedhypersurface Σ ⊂ R n +1 the value(1) A Φ (Σ) := (cid:90) Σ Φ( ν ) dA, where ν is the unit normal to Σ, and Φ is a one-homogeneous function on R n +1 that is positive and C on S n , and has uniformly convex sub-level sets. Functionalsof the form (1) have attracted recent attention for their applied and theoreticalinterest ([6], [7]). In particular, they arise in models of crystal surfaces and inFinsler geometry, and the lack of a monotonicity formula for critical points of (1)presents interesting technical challenges. Almgren-Schoen-Simon proved that the( n − n = 2. Morganlater showed that the cone over S k × S k in R k +2 minimizes a parametric ellipticfunctional for each k ≥
1, by constructing a calibration [14]. Thus, there existsingular minimizers of parametric elliptic functionals when n ≥ S k × S l minimize para-metric elliptic functionals for each k, l ≥
1, by constructing foliations by minimizersin the spirit of [3]. To our knowledge, this is the first application of the foliationapproach for integrands other than area (that is, Φ | S n = 1). Mathematics Subject Classification.
Key words and phrases.
Elliptic integrands, singular minimizers, Bernstein problem. a r X i v : . [ m a t h . A P ] F e b CONNOR MOONEY AND YANG YANG
Remark . The examples here and in [14] show that best regularity result pos-sible for minimizers of (1) is that the singular set has e.g. locally bounded n − u , we say that u solves an equationof minimal surface type. An interesting question is: Question 1.2.
Are entire solutions to equations of minimal surface type in R n necessarily linear? For the area functional, the answer to Question 1.2 is “yes” if n ≤ n ≥ n ≤ n ≥ n = 4 , n ≥ k, l ≥
1, we let x ∈ R k +1 and y ∈ R l +1 , and we define the Lawson cones C kl over S k × S l by C kl := {| x | = | y |} ⊂ R k + l +2 . Theorem 1.3.
For each k, l ≥ , there exist parametric elliptic functionals A Φ such that Φ is analytic away from the origin, and each side of the cone C kl isfoliated by analytic minimizers of A Φ . In particular, C kl minimizes A Φ .Remark . The foliation is generated by dilations of a pair of critical points of A Φ , each of which lies on one side of C kl and is asymptotic to C kl at infinity. Wediscuss the precise asymptotic behavior in Section 3. Remark . The A Φ -minimality of C kl seems to be new in the cases k (cid:54) = l and k + l ≤
5, or k + l = 6 and min { k, l } = 1. The cases k = l ≤ k + l = 6 and min { k, l } ≥ k + l ≥
7) are knownto minimize area, up to making an affine transformation ([12], [15]).
Remark . A proof of minimality by foliation gives rise to a proof by calibrationthrough the following observation: If we denote by ν ( z ) the unit normal to the leafthat passes through z , then the vector field ∇ Φ( ν ) is a calibration on R k + l +2 . Fora discussion of this connection in the area case see e.g. [4].The paper is organized as follows. In Section 2 we prove Theorem 1.3. We reducethe problem to the careful analysis of a certain nonlinear second-order ODE, usingthe symmetries of C kl . The heart of our construction is Lemma 2.1, which givesconditions on the integrand that guarantee the existence of solutions to this ODEwith the desired properties. In Section 3 we discuss the behavior at infinity of the OLIATION 3 leaves in the foliation given by Theorem 1.3, and the implications for Question 1.2.In particular, we state conjectures concerning the existence and growth rates ofnonlinear global solutions to equations of minimal surface type in R k + l +2 for each k, l ≥
1, and we compare these conjectures with what is known about the minimalsurface equation.
Acknowledgements
This research was supported by NSF grant DMS-1854788.2.
Proof of Theorem 1.3
Integrand Notation.
We choose integrands Φ that depend only on | x | and | y | . We define them by a pair of one-variable functions φ and ψ as follows:(2) Φ( x, y ) = | y | φ (cid:16) | x || y | (cid:17) , | y | ≥ | x || x | ψ (cid:16) | y || x | (cid:17) , | x | > | y | . The functions φ and ψ will be chosen to be positive, smooth, even, and locallyuniformly convex on R .2.2. Foliation Leaf Notation.
Having fixed an appropriate choice of Φ, we willshow that there exists a critical point of A Φ of the form(3) Σ kl = {| y | = σ ( | x | ) } ⊂ {| y | > | x |} , where σ is smooth, even, convex, asymptotic to | . | , and | σ (cid:48) | < . The dilations ofΣ kl are then minimizers of A Φ , and they foliate one side of C kl (namely {| y | > | x |} ),see Figure 1. A similar procedure will give a foliation of the other side {| x | > | y |} by minimizers of A Φ .2.3. Euler-Lagrange ODE.
For hypersurfaces of the form (3) and integrands ofthe form (2), the condition that Σ kl is a critical point of A Φ is equivalent to thenonlinear second-order ODE(4) σ (cid:48)(cid:48) ( t ) + kP ( σ (cid:48) ( t )) σ (cid:48) ( t ) t + lQ ( σ (cid:48) ( t )) 1 σ ( t ) = 0 , where(5) P ( s ) := φ (cid:48) ( s ) sφ (cid:48)(cid:48) ( s ) , Q ( s ) := sφ (cid:48) ( s ) − φ ( s ) φ (cid:48)(cid:48) ( s ) . This follows from the first variation formula tr ( D Φ( ν ( z )) II ( z )) = 0for critical points of (1), where II denotes the second fundamental form of Σ and ν denotes the unit normal. One can also use the symmetries of Σ kl and Φ to reducethe problem to taking the first variation of the one-variable integral A Φ (Σ kl ) = const. (cid:90) t k σ l ( t ) φ ( σ (cid:48) ( t )) dt. In the following technical lemma we show that there exists a global solution to(4) with the desired properties, provided φ satisfies certain analytic conditions. We CONNOR MOONEY AND YANG YANG ℝ k +1 ℝ l +1 C kl Σ kl x . σ ( | x | ) Figure 1.
The dilations of Σ kl foliate one side of C kl .will later give examples of φ that satisfy these conditions. To state the lemma wedefine for a smooth function ϕ on R the function E kl ( ϕ ) by E kl ( ϕ )( s ) := l k + l − k + l + 1 ϕ ( s ) − (cid:18) k + (cid:18) l − k + lk + l + 1 (cid:19) s (cid:19) ϕ (cid:48) ( s ) − (cid:18) k + l + 12 − s (cid:19) (1 − s ) ϕ (cid:48)(cid:48) ( s ) . (6) Lemma 2.1.
Assume that φ ( s ) is a smooth, even, uniformly convex function on R that satisfies (7) φ (1) = 1 , φ (cid:48) (1) = lk + l , and in addition that (8) E kl ( φ )( s ) ≥ κ (1 − s ) for some κ > and all s ∈ [0 , . Then there exists a global smooth, even, convexsolution σ to the ODE (4) that satisfies the initial conditions (9) σ (0) = 1 , σ (cid:48) (0) = 0 OLIATION 5 and in addition satisfies σ ( t ) > | t | , | σ (cid:48) ( t ) | < for all t , and (10) σ ( t ) = | t | + a | t | − µ + o ( | t | − µ ) as t → ∞ for some a > , where (11) µ := k + l − − (cid:115)(cid:18) k + l − (cid:19) − klφ (cid:48)(cid:48) (1)( k + l ) . Remark . It is straightforward to check that any function φ satisfying the con-ditions (7) and (8) automatically satisfies the inequality(12) φ (cid:48)(cid:48) (1) − kl ( k + l )( k + l − > , so µ is well-defined. Conversely, any choice of φ that satisfies (7) and (12) alsosatisfies (8) for s ∈ [1 − δ, κ > δ > (cid:107) φ (cid:107) C ([ − , . Proof of Lemma 2.1.
Standard ODE theory gives the short-time existence of asolution to (4) with the desired properties in a neighborhood of 0 (see Remark 2.3below). To proceed we rewrite (4) as an autonomous first-order system. In termsof the quantities(13) w ( τ ) := e − τ σ ( e τ ) , z ( τ ) := σ (cid:48) ( e τ ) , the second-order ODE (4) becomes:(14) (cid:18) w (cid:48) z (cid:48) (cid:19) = (cid:18) − w + z − l Q ( z ) w − kzP ( z ) (cid:19) := V ( w, z ) . We denote the components of the vector field V by V i , i = 1 ,
2, and the solutioncurve ( w ( τ ) , z ( τ )) by Γ( τ ). The only zero of V in the infinite half-stripΩ := { w ≥ } ∩ { ≤ z ≤ } occurs at (1 , ,
1) has the form X (cid:48) = M X ,where(15) M = (cid:18) − − klφ (cid:48)(cid:48) (1)( k + l ) − k − l (cid:19) . The eigenvalues of M are(16) λ ± = − k + l + 12 ± (cid:115)(cid:18) k + l − (cid:19) − klφ (cid:48)(cid:48) (1)( k + l ) , and these eigenvalues correspond to directions with slopes 1 + λ ± .We claim that Γ is contained in the region R ⊂ Ω bounded by the curvesΓ := { z = 0 } , Γ := { V = 0 } = (cid:26) w = lk (cid:18) φφ (cid:48) − z (cid:19)(cid:27) , andΓ := (cid:26) ( z −
1) = (cid:18) λ + + λ − (cid:19) ( w − (cid:27) (see Figure 2). Since R ⊂ Ω ∩ { V > } , this would imply that σ > | . | , | σ (cid:48) | < σ is convex.The inclusion Γ( τ ) ∈ R when τ << σ , so we justneed to check that V points towards the interior of R on each of the curves { Γ i } i =1 . CONNOR MOONEY AND YANG YANG Γ Γ Γ R z = 1 w = 1 Figure 2.
The solution curve is contained in the region R bounded by Γ , Γ and Γ .This holds on Γ using that Q (0) <
0. The uniform convexity of φ guarantees thatΓ is a graph over the positive z -axis with negative slope, and since V < . Finally, inequality (8) guarantees that thedesired geometric condition holds on Γ , after a calculation (which we omit) usingthe definition (5) of the quantities P and Q .The asymptotic behavior (10) then follows from the linear analysis. Indeed, theregion R excludes the line with slope 1 + λ − that goes through (1 , τ large, the coefficient of the principal eigenvector of thelinearized operator at (1 ,
1) (which corresponds to the eigenvalue λ + and has slope − µ = 1 + λ + ) is nonzero, completing the proof. (cid:3) Remark . We could not find a precise reference for short-time existence, so forcompleteness we sketch the argument. We first rewrite (4) in divergence form:[ t k σ l φ (cid:48) ( σ (cid:48) )] (cid:48) = lt k σ l − φ ( σ (cid:48) ) . We are thus looking for a continuous function σ (cid:48) on an interval [0 , t ] such that σ (cid:48) (0) = 0 and σ (cid:48) ( t ) = ( φ ∗ ) (cid:48) lt k (cid:16) (cid:82) t σ (cid:48) ( s ) ds (cid:17) l (cid:90) t s k (cid:18) (cid:90) s σ (cid:48) ( τ ) dτ (cid:19) l − φ ( σ (cid:48) ( s )) ds := G ( σ (cid:48) )( t ) , OLIATION 7 where φ ∗ is the Legendre transform of φ . For t > G is acontraction mapping on the space of continuous functions on [0 , t ] that vanishat 0 and are bounded by 1 in the C norm. A fixed point argument then givesthe existence of a function σ ∈ C [0 , t ] ∩ C ∞ (0 , t ) that solves (4) on (0 , t ) andsatisfies σ (0) = 1 , σ (cid:48) (0) = 0. The higher regularity of σ follows from the observationthat Σ kl = {| y | = σ ( | x | ) } can be locally written over its tangent planes as a C graph that solves an equation of minimal surface type (and is thus smooth, see e.g.[10]). Finally, the equation (4) itself gives that σ (cid:48)(cid:48) (0) = lφ (0)( k + 1) φ (cid:48)(cid:48) (0) > , so σ is convex near 0, concluding the argument.2.4. Proof of Main Theorem.
In this final subsection we choose the functions φ, ψ that define Φ, and we apply Lemma 2.1 to prove Theorem 1.3.
Proof of Theorem 1.3.
We first indicate how to choose integrands Φ that are C , away from the origin, defined through the notation (2). For p, q > φ ( s ) = 1 − lp ( k + l ) + lp ( k + l ) | s | p ,ψ ( s ) = 1 − kq ( k + l ) + kq ( k + l ) | s | q , (17)up to making small perturbations near s = 0 so that φ and ψ are smooth anduniformly convex. It is straightforward to check that if p and q are related by(18) l ( p −
1) = k ( q − , then Φ is C , away from the origin. We note that φ satisfies (7), and we will verifythat provided p is sufficiently large, then φ also satisfies the desired inequality(8). Away from a small neighborhood of s = 0, where we perturbed (17) and theinequality (8) is obvious, the inequality E kl ( φ )( s ) > (cid:20) ( p − (cid:18) k + l + 12 − s (cid:19) + ks (cid:21) (1 − s ) < ( k + l − k + l − l/p ) k + l + 1 ( s − p − s ) . (19)Denote the left side of (19) by L ( s ) and the right side by R ( s ). Since L is quadraticin s and R (cid:48)(cid:48) is decreasing in s , it suffices to prove the inequalities R (cid:48) (1) < L (cid:48) (1) , L (cid:48)(cid:48) ≤ R (cid:48)(cid:48) (1) . The first inequality holds provided(20) p − > k ( k + l − , in agreement with Remark (2.2). The second one holds provided( p − − (cid:18) k + 2 l + 2 k + l + 4( k + l )( k + l − (cid:19) ( p − k ( k + l )( k + l − ≥ . (21) CONNOR MOONEY AND YANG YANG
Both (20) and (21) hold e.g. when p ≥
6, regardless of k, l ≥
1. When p ≥ κ > k and l , the function ψ satisfies (7), and a similar analysis showsthat E lk ( ψ ) ≥ κ (1 − s ) for some κ > s ∈ [0 ,
1] if q ≥
6. We concludeusing Lemma 2.1 that each side of C kl is foliated by smooth critical points of A Φ when we choose φ, ψ as above with p, q ≥
6, and furthermore Φ ∈ C , ( S k + l +1 )provided p and q are chosen such that (18) holds as well.We now explain how the integrand can be made analytic on S k + l +1 , by perturbingthe C , integrand constructed above. We first improve to smooth. Take φ and ψ as above, and let ˜ φ ( s ) = sψ (1 /s )for s ≤
1. We glue φ to ˜ φ near s = 1 by taking the convex combination¯ φ := η δ φ + (1 − η δ ) ˜ φ, where η δ is a smooth function that transitions from 1 to 0 in the interval [1 − δ, − δ ]for δ > (cid:107) η δ (cid:107) C m ( R ) ≤ C m δ − m with C m independent of δ . Since ˜ φ and φ agree to second order at s = 1, theinequality (8) holds for ¯ φ away from [1 − δ, − δ ] provided δ is small (see Remark(2.2)). Furthermore, we have | ˜ φ ( m ) ( s ) − φ ( m ) ( s ) | ≤ C m (1 − s ) − m for each m ≤ s ∈ [0 , | E kl ( ¯ φ ) − E kl ( φ ) | ≤ Cδ in [1 − δ, − δ ]. Since E kl ( φ ) ≥ κδ in this interval, the inequality (8) holds for ¯ φ when δ is small, up to reducing κ slightly. After replacing φ by ¯ φ (and keeping ψ the same), we obtain a new integrand that is smooth on S k + l +1 and by Lemma 2.1satisfies the desired properties.Finally, we indicate how to improve the regularity from smooth to analytic.We start with a smooth choice of integrand Φ as constructed above. Using thesymmetries of Φ we may view it as a smooth function on S . We approximatethis function by the partial sums S N of its Fourier series with N terms. We addsmall correctors of the form a N + b N cos(2 θ ) + c N cos(4 θ ) to S N to obtain newapproximations T N , with a N , b N , c N chosen such that T N agrees to second orderwith Φ at θ = π/
4. Since S N converge uniformly in C m to Φ for any m , thefunctions T N do as well. It follows that the one-homogeneous extensions of T N to R (which we now identify with T N ) have uniformly convex sub-level sets for N large. Since T N agree to second order with Φ on the diagonals, Remark (2.2)implies that the conditions (7) and (8) hold for the function obtained by restricting T N to the horizontal lines tangent to S when N is large. The same holds (with k and l exchanged) for the restriction of T N to the vertical lines tangent to S . Hence,after replacing Φ( x, y ) with T N ( | x | , | y | ) for N large, we obtain an integrand thatis analytic on S k + l +1 and by Lemma 2.1 satisfies the desired properties. (cid:3) OLIATION 9 Discussion
In this section we discuss the implications of the analysis in Section 2 for Ques-tion 1.2. The discussion is motivated by the examples of entire minimal graphsconstructed in [3] and [16]. Those examples are asymptotic to area-minimizingcones of the form K × R , where K is the Simons cone in [3], and any one of a largefamily of area-minimizing cones with isolated singularities in [16]. In all cases, eachside of K is foliated by smooth area-minimizing hypersurfaces. These are closelyrelated to the level sets of the functions u that define the entire minimal graphs.More precisely, each level set of u is a graph over K outside of some ball, withthe same leading-order asymptotic behavior at infinity as a leaf in the foliation.Furthermore, if the distance between a leaf in the foliation and K on ∂B r behaveslike r − µ as r → ∞ , then sup B r |∇ u | ∼ r µ .In view of this discussion we conjecture: Conjecture 3.1.
For any integrand Φ as constructed in Theorem 1.3, there existsan elliptic extension of Φ to R k + l +3 , and a nonlinear global solution to the corre-sponding equation of minimal surface type in R k + l +2 , whose graph is asymptotic to C kl × R . Moreover, the gradient of this solution grows at the same rate that theleaves in the foliation associated to Φ approach C kl . The proof of Theorem 1.3 shows that for any µ ∈ (0 , µ kl ), we can choose integrandssuch that each side of C kl is foliated by minimizers whose distance from C kl on ∂B r behaves like r − µ , where(22) µ kl = k + l − − (cid:115)(cid:18) k + l − (cid:19) − min { k, l } . The formula (22) comes from (11) and noting that, when choosing φ and ψ , we couldtake any exponents p and q such that (18) holds and p, q ≥
6. Thus, Conjecture(3.1) predicts that for any µ ∈ (0 , µ kl ), there exist global solutions to equations ofminimal surface type in R k + l +2 whose graphs are are asymptotic to C kl × R , andhave maximum gradient in B r growing like r µ . Remark . The first author showed in [13] that when k = l = 2, the graph of u = | x | − | y | (which is asymptotic to C × R ) minimizes a parametric ellipticfunctional A Ψ , and each level set of u minimizes A Φ , where Φ = Ψ | { x =0 } . Theperspective in that work is quite different, and the proof is based on solving alinear hyperbolic equation to construct Ψ. However, the discussion at the end of[13] shows that this strategy could be challenging to implement when 2 ≤ k + l ≤ k = l = 1, a closer inspection of inequalities (20) and (21)shows that we can take any p = q > φ and ψ . This correspondsto the “optimal” value µ = . One may hope to show that (22) can be improvedto µ kl = ( k + l − / k and l , which corresponds to choices of φ such that inequality (12) tends to equality. However, we suspect that this is notpossible. Indeed, when k + l is large, this corresponds to a small value of φ (cid:48)(cid:48) (1).It would follow that the integrand Φ is larger on S k × S l than at nearby points on √ S k + l +1 , in which case perturbations of C kl could likely decrease its energy.Since the quantity (22) is bounded above independently of k and l , it does not seemlikely that the examples from Theorem 1.3 can give rise to solutions to equationsof minimal surface type with arbitrarily fast gradient growth.On the other hand, Conjecture 3.1 predicts the existence of global solutions toequations of minimal surface type with very slow gradient growth, namelysup B r |∇ u | ∼ r µ with µ > µ tends to zero, and we conjecture a “quantitative” versionof the rigidity result for solutions with bounded gradient: Conjecture 3.3.
Let u be a global solution to an equation of minimal surface typeon R n , corresponding to a functional A Ψ . Then for some (cid:15) ( n, Ψ) > , sup B r |∇ u | = O ( r (cid:15) ) ⇒ u is linear. In [8] the authors give a beautiful proof of Conjecture 3.3 for the area functional,for any (cid:15) < n . The proof in [8] depends on preciseconstants in the Simons inequality for the Laplacian of the second fundamentalform on a minimal surface. Although analogues of the Simons inequality exist forcritical points of (1), the constants degenerate with the ellipticity of Φ, and it isnot clear that the same strategy would prove Conjecture 3.3. References [1] Almgren, Jr., F. J. Some interior regularity theorems for minimal surfaces and an extensionof Bernstein’s theorem.
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OLIATION 11 [13] Mooney, C. Entire solutions to equations of minimal surface type in six dimensions.
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Ann. of Math. (1968), 62-105. Department of Mathematics, UC Irvine
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