<p>We study the global existence and stability of surface diffusion flow (the normal velocity is given by the Laplacian of the mean curvature) of smooth boundaries of subsets of the <i>n</i>–dimensional flat torus. More precisely, we show that if a smooth set is “close enough” to a strictly stable critical set for the Area functional under a volume constraint, then the surface diffusion flow of its boundary hypersurface exists for all time and asymptotically converges to the boundary of a “translated” of the critical set. This result was obtained in dimension <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n=3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> by Acerbi, Fusco, Julin and Morini in [<CitationRef CitationID="CR1">1</CitationRef>] (extending previous results for spheres of Escher, Mayer and Simonett [<CitationRef CitationID="CR15">15</CitationRef>], Wheeler [<CitationRef CitationID="CR37">37</CitationRef>, <CitationRef CitationID="CR38">38</CitationRef>] and Elliott and Garcke [<CitationRef CitationID="CR14">14</CitationRef>]). Our work generalizes such conclusion to any dimension <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\in \mathbb {N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>. For sake of clarity, we show all the details in dimension <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n=4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> and we list the necessary modifications to the quantities involved in the proof in the general <i>n</i>–dimensional case, in the last section.</p>