MAnET Metric Analysis Meeting 18 – 19 January, 2017 A new stable surface in the Heisenberg group Fellow: Sebastiano Nicolussi Golo Advisors: Francesco Serra Cassano Enrico Le Donne
Introduction The Heisenberg group, H : three dimensional, simply connected, nilpotent, non-Abelian Lie group. In exponential coordinates, we identify H = R 3 . ◮ Group operation � a + x, b + y, c + z + 1 � ( a, b, c ) ∗ ( x, y, z ) = 2( ay − bx ) , ◮ Left-invariant vector fields X ( x, y, z ) = ∂ x − 1 2 y∂ z Y ( x, y, z ) = ∂ y + 1 2 x∂ z Z ( x, y, z ) = ∂ z ,
Introduction ◮ Intrinsic graph of f : R 2 → R : � (0 , η, τ ) ∗ ( f ( η, τ ) , 0 , 0) : ( η, τ ) ∈ R 2 � Γ f = � ( f, η, τ − 1 � 2 ηf ) : ( η, τ ) ∈ R 2 = , ◮ Intrinsic derivative of f ∇ f f = ∂ η f + f∂ τ f, which describes the tangent space of Γ f , W is the set of all C 1 -intrinsic functions , i.e., f : R 2 → R ◮ C 1 such that both f and ∇ f f are continuous. ◮ Intrinsic area of Γ f over ω ⊂ R 2 � � 1 + ( ∇ f f ) 2 d η d τ. ω
Introduction Remarks ◮ There is a natural notion of subRiemannian perimeter for subsets E ⊂ H . If E has locally finite perimeter, then its reduced boundary ∂ ∗ E is “essentially” the intrinsic graph of a C 1 -intrinsic function.
Introduction Remarks ◮ There is a natural notion of subRiemannian perimeter for subsets E ⊂ H . If E has locally finite perimeter, then its reduced boundary ∂ ∗ E is “essentially” the intrinsic graph of a C 1 -intrinsic function. ◮ There are C 1 -intrinsic functions f ∈ C 1 W whose intrinsic graph Γ f is a fractal in R 3 .
Introduction Remarks ◮ There is a natural notion of subRiemannian perimeter for subsets E ⊂ H . If E has locally finite perimeter, then its reduced boundary ∂ ∗ E is “essentially” the intrinsic graph of a C 1 -intrinsic function. ◮ There are C 1 -intrinsic functions f ∈ C 1 W whose intrinsic graph Γ f is a fractal in R 3 . ◮ The set C 1 W is NOT a vector space. Indeed, f ∈ C 1 W �⇒ f + 1 ∈ BV intrinsic,loc
Introduction Remarks ◮ There is a natural notion of subRiemannian perimeter for subsets E ⊂ H . If E has locally finite perimeter, then its reduced boundary ∂ ∗ E is “essentially” the intrinsic graph of a C 1 -intrinsic function. ◮ There are C 1 -intrinsic functions f ∈ C 1 W whose intrinsic graph Γ f is a fractal in R 3 . ◮ The set C 1 W is NOT a vector space. Indeed, f ∈ C 1 W �⇒ f + 1 ∈ BV intrinsic,loc ◮ By minimal surface in H we mean a topological surface that minimizes the area among all its bounded variations.
The problem Bernstein’s Problem : Under which conditions on f is the following sentence true? If Γ f is a minimal surface in H , then Γ f is a vertical plane. We are interested in this problem because we want to better understand the space C 1 W and the theory of perimeters in H . ◮ If f ∈ C 1 ( R 2 ) , then it’s true! 1 ◮ If f ∈ C 0 ( R 2 ) (even Lipschitz intrinsic), then it’s false! 2 1 M. Galli and M. Ritoré. “Area-stationary and stable surfaces of class C 1 in the sub-Riemannian Heisenberg group H 1 ”. In: Adv. Math. 285 (2015), pp. 737–765. 2 R. Monti, F. Serra Cassano, and D. Vittone. “A negative answer to the Bernstein problem for intrinsic graphs in the Heisenberg group”. In: Boll. Unione Mat. Ital. (9) 1.3 (2008), pp. 709–727.
The problem Bernstein’s Problem : Under which conditions on f is the following sentence true? If Γ f is a minimal surface in H , then Γ f is a vertical plane. We are interested in this problem because we want to better understand the space C 1 W and the theory of perimeters in H . ◮ If f ∈ C 1 ( R 2 ) , then it’s true! 1 ◮ If f ∈ C 0 ( R 2 ) (even Lipschitz intrinsic), then it’s false! 2 ◮ What if f ∈ C 1 W ? 1 M. Galli and M. Ritoré. “Area-stationary and stable surfaces of class C 1 in the sub-Riemannian Heisenberg group H 1 ”. In: Adv. Math. 285 (2015), pp. 737–765. 2 R. Monti, F. Serra Cassano, and D. Vittone. “A negative answer to the Bernstein problem for intrinsic graphs in the Heisenberg group”. In: Boll. Unione Mat. Ital. (9) 1.3 (2008), pp. 709–727.
The problem Variational approach: if Γ f is a minimal surface, then, for all ψ ∈ C ∞ c ( ω ) , the following two conditions are satisfied: � 1 + ( ∇ f + ǫψ ( f + ǫψ )) 2 d η d τ � d � 1. I f ( ψ ) = ǫ =0 = 0 � d ǫ ω � � 1 + ( ∇ f + ǫψ ( f + ǫψ )) 2 d η d τ d 2 � � 2. II f ( ψ ) = ǫ =0 ≥ 0 � d ǫ 2 ω �
The problem Variational approach: if Γ f is a minimal surface, then, for all ψ ∈ C ∞ c ( ω ) , the following two conditions are satisfied: � 1 + ( ∇ f + ǫψ ( f + ǫψ )) 2 d η d τ � d � 1. I f ( ψ ) = ǫ =0 = 0 � d ǫ ω � � 1 + ( ∇ f + ǫψ ( f + ǫψ )) 2 d η d τ d 2 � � 2. II f ( ψ ) = ǫ =0 ≥ 0 � d ǫ 2 ω � This approach cannot work in general: If we only assume f ∈ C 1 W , then it is not true that the graph of the function f + ǫψ has necessarily locally finite area! Indeed, f ∈ C 1 W �⇒ f + 1 ∈ BV intrinsic,loc
The problem Variational approach: if Γ f is a minimal surface, then, for all ψ ∈ C ∞ c ( ω ) , the following two conditions are satisfied: � 1 + ( ∇ f + ǫψ ( f + ǫψ )) 2 d η d τ � d � 1. I f ( ψ ) = ǫ =0 = 0 � d ǫ ω � � 1 + ( ∇ f + ǫψ ( f + ǫψ )) 2 d η d τ d 2 � � 2. II f ( ψ ) = ǫ =0 ≥ 0 � d ǫ 2 ω � This approach cannot work in general: If we only assume f ∈ C 1 W , then it is not true that the graph of the function f + ǫψ has necessarily locally finite area! Indeed, f ∈ C 1 W �⇒ f + 1 ∈ BV intrinsic,loc But this is another story...
The problem Variational approach: if Γ f is a minimal surface, then, for all ψ ∈ C ∞ c ( ω ) , the following two conditions are satisfied: � 1 + ( ∇ f + ǫψ ( f + ǫψ )) 2 d η d τ � d � 1. I f ( ψ ) = ǫ =0 = 0 � d ǫ ω � � 1 + ( ∇ f + ǫψ ( f + ǫψ )) 2 d η d τ d 2 � � 2. II f ( ψ ) = ǫ =0 ≥ 0 � d ǫ 2 ω � This approach cannot work in general: If we only assume f ∈ C 1 W , then it is not true that the graph of the function f + ǫψ has necessarily locally finite area! Indeed, f ∈ C 1 W �⇒ f + 1 ∈ BV intrinsic,loc But this is another story... We decided tackle the following problem: If f ∈ W 1 , 1 loc ( R 2 ) ∩ C 1 W and Γ f is a minimal surface in H , then Γ f is a vertical plane.
First Variation For f ∈ W 1 , 1 loc ( R 2 ) ∩ C 1 c ( R 2 ) , W and ψ ∈ C ∞ ∇ f f � I f ( ψ ) = 1 + ( ∇ f f ) 2 ( ∂ η ψ + ∂ τ ( fψ )) d η d τ. � R 2
First Variation For f ∈ W 1 , 1 loc ( R 2 ) ∩ C 1 c ( R 2 ) , W and ψ ∈ C ∞ ∇ f f � I f ( ψ ) = 1 + ( ∇ f f ) 2 ( ∂ η ψ + ∂ τ ( fψ )) d η d τ. � R 2 If f ∈ C 2 ( R 2 ) , then � � ∇ f f � R 2 ∇ f I f ( ψ ) = − ψ d η d τ. � 1 + ( ∇ f f ) 2
First Variation For f ∈ W 1 , 1 loc ( R 2 ) ∩ C 1 c ( R 2 ) , W and ψ ∈ C ∞ ∇ f f � I f ( ψ ) = 1 + ( ∇ f f ) 2 ( ∂ η ψ + ∂ τ ( fψ )) d η d τ. � R 2 If f ∈ C 2 ( R 2 ) , then � � ∇ f f � R 2 ∇ f I f ( ψ ) = − ψ d η d τ. � 1 + ( ∇ f f ) 2 Thus, I f = 0 if and only if ∇ f ∇ f f = 0 . This is a second order differential equation.
First Variation For f ∈ W 1 , 1 loc ( R 2 ) ∩ C 1 c ( R 2 ) , W and ψ ∈ C ∞ ∇ f f � I f ( ψ ) = 1 + ( ∇ f f ) 2 ( ∂ η ψ + ∂ τ ( fψ )) d η d τ. � R 2 If f ∈ C 2 ( R 2 ) , then � � ∇ f f � R 2 ∇ f I f ( ψ ) = − ψ d η d τ. � 1 + ( ∇ f f ) 2 Thus, I f = 0 if and only if ∇ f ∇ f f = 0 . This is a second order differential equation. However, we can interpret it as ( Lagrangian interpretation ) ∇ f f is constant along the integral curves of the vector field ∇ f = ∂ η + f∂ τ .
First Variation Conjecture If f ∈ W 1 , 1 loc ∩ C 1 W and I f = 0 , then ∇ f f is constant along the integral lines of ∇ f , i.e., ∇ f ∇ f f = 0 . We are working on it. However, we know that, if f ∈ W 1 , 1 W and ∇ f f is constant loc ∩ C 1 along the integral lines of ∇ f , i.e., ∇ f ∇ f f = 0 , then I f = 0 .
∇ f ∇ f f = 0 ∇ f = ∂ η + f∂ τ Lemma W . ∇ f f is constant along the integral curves of ∇ f if Let f ∈ C 1 and only if there are continuous functions A, B : R → R such that 1. all the integral curves of ∇ f are given by t �→ ( t, g ( t, ζ )) , where g ( t, ζ ) = A ( ζ ) t 2 + B ( ζ ) t + ζ ; 2
∇ f ∇ f f = 0 ∇ f = ∂ η + f∂ τ Lemma W . ∇ f f is constant along the integral curves of ∇ f if Let f ∈ C 1 and only if there are continuous functions A, B : R → R such that 1. all the integral curves of ∇ f are given by t �→ ( t, g ( t, ζ )) , where g ( t, ζ ) = A ( ζ ) t 2 + B ( ζ ) t + ζ ; 2 2. The functions A, B must satisfy some properties. In particular, A must be non-decreasing;
∇ f ∇ f f = 0 ∇ f = ∂ η + f∂ τ Lemma W . ∇ f f is constant along the integral curves of ∇ f if Let f ∈ C 1 and only if there are continuous functions A, B : R → R such that 1. all the integral curves of ∇ f are given by t �→ ( t, g ( t, ζ )) , where g ( t, ζ ) = A ( ζ ) t 2 + B ( ζ ) t + ζ ; 2 2. The functions A, B must satisfy some properties. In particular, A must be non-decreasing; 3. f ( t, g ( t, ζ )) = ∂ t g ( t, z ) = A ( ζ ) t + B ( ζ ); 4. ∇ f f ( t, g ( t, ζ )) = ∂ 2 t g ( t, z ) = A ( ζ ) .
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