Simons cone

<h2 id="definition">Definition</h2>
<p>The Simons cone is defined as the <a href="/facts/Hypersurface/cTvtLVsV">hypersurface</a> given by the equation
</p>

S
        =
        {
        x
        ∈
        
          
            R
          
          
            8
          
        
        
          |
        
        
          x
          
            1
          
          
            2
          
        
        +
        
          x
          
            2
          
          
            2
          
        
        +
        
          x
          
            3
          
          
            2
          
        
        +
        
          x
          
            4
          
          
            2
          
        
        =
        
          x
          
            5
          
          
            2
          
        
        +
        
          x
          
            6
          
          
            2
          
        
        +
        
          x
          
            7
          
          
            2
          
        
        +
        
          x
          
            8
          
          
            2
          
        
        }
        ⊂
        
          
            R
          
          
            8
          
        
      
    
    {\displaystyle S=\{x\in \mathbb {R} ^{8}|x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\}\subset \mathbb {R} ^{8}}
  
.
<p>This 7-dimensional <a href="/facts/Cone_(mathematics)/2c9GIzYD">cone</a> has the distinctive property that its <a href="/facts/Mean_curvature/lgdJ9b0k">mean curvature</a> vanishes at every point except at the origin, where the cone has a <a href="/facts/Singularity_(mathematics)/Y5sMQY4I">singularity</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="applications">Applications</h2>
<p class="note">See also: <a href="/facts/Bernstein%2527s_problem/FPBA1fcp">Bernstein's problem</a></p>
<p>The classical Bernstein theorem states that any minimal graph in 
  
    
      
        
          
            R
          
          
            3
          
        
      
    
    {\displaystyle \mathbb {R} ^{3}}
  
 must be a plane. This was extended to 
  
    
      
        
          
            R
          
          
            4
          
        
      
    
    {\displaystyle \mathbb {R} ^{4}}
  
 by <a href="/facts/Wendell_Fleming/6uAMYpuD">Wendell Fleming</a> in 1962 and <a href="/facts/Ennio_De_Giorgi/B99fBWmf">Ennio De Giorgi</a> in 1965, and to dimensions up to 
  
    
      
        
          
            R
          
          
            5
          
        
      
    
    {\displaystyle \mathbb {R} ^{5}}
  
 by <a href="/facts/Frederick_J._Almgren_Jr./t3QjbN0v">Frederick J. Almgren Jr.</a> in 1966 and to 
  
    
      
        
          
            R
          
          
            8
          
        
      
    
    {\displaystyle \mathbb {R} ^{8}}
  
 by <a href="/facts/Jim_Simons/op947h9w">Jim Simons</a> in 1968. The existence of the Simons cone as a minimizing cone in 
  
    
      
        
          
            R
          
          
            8
          
        
      
    
    {\displaystyle \mathbb {R} ^{8}}
  
 demonstrated that the Bernstein theorem could not be extended to 
  
    
      
        
          
            R
          
          
            9
          
        
      
    
    {\displaystyle \mathbb {R} ^{9}}
  
 and higher dimensions. Bombieri, De Giorgi, and <a href="/facts/Enrico_Giusti/JP6une46">Enrico Giusti</a> proved in 1969 that the Simons cone is indeed area-minimizing, thus providing a negative answer to the Bernstein problem in higher dimensions.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Minimal_surface/Fy7VkrLQ">Minimal surface</a></li>
<li><a href="/facts/Bernstein%2527s_problem/FPBA1fcp">Bernstein's problem</a></li>
<li><a href="/facts/Geometric_measure_theory/NLIQxYf9">Geometric measure theory</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Bombieri, E., De Giorgi, E., and Giusti, E. (1969). "Minimal cones and the Bernstein problem". Inventiones Mathematicae, 7: 243-268. <a href="/wiki/Inventiones_Mathematicae" target="_blank">/wiki/Inventiones_Mathematicae</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>G. De Philippis, E. Paolini (2009). "A short proof of the minimality of Simons cone". Rendiconti del Seminario Matematico della Università di Padova, 121. pp. 233-241 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Bombieri, E., De Giorgi, E., and Giusti, E. (1969). "Minimal cones and the Bernstein problem". Inventiones Mathematicae, 7: 243-268. <a href="/wiki/Inventiones_Mathematicae" target="_blank">/wiki/Inventiones_Mathematicae</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>G. De Philippis, E. Paolini (2009). "A short proof of the minimality of Simons cone". Rendiconti del Seminario Matematico della Università di Padova, 121. pp. 233-241 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Simons cone open-in-new

Simons cone